EUROPEAN SCHOOL FOR ADVANCED STUDIES IN
REDUCTION OF SEISMIC RISK
ROSE SCHOOL
DESIGN VERIFICATION OF A
DISPLACEMENT-BASED DESIGNED
THREE DIMENSIONAL WALL-STEEL FRAME
BUILDING
A Dissertation Submitted in Partial Fulfilment
of the Requirements for the Master Degree in
EARTHQUAKE ENGINNERING
by
CLARA CAPONI
Supervisor: Dr RUI PINHO
May, 2008
The dissertation entitled “Design verification of a displacement-based designed three
dimensional wall-steel frame building”, by Clara Caponi, has been approved in partial
fulfilment of the requirements for the Master Degree in Earthquake Engineering.
Dr. Rui Pinho …… …
………
Dr. T.J. Sullivan………… …
……
Abstract
ABSTRACT
Nonlinear dynamic time-history analyses are carried out in order to verify the efficiency and
consistency of DDBD design techniques focused on dual frame-wall structure. Following this
aim, a three dimensional prototype structure consisting of two-way moment resisting
structural steel frames with channel walls of reinforced concrete was entirely defined and
examined. From the first steps of the procedure, a complete design of the building was
performed considering the flexural strength requirements indicated by the DDBD design
method.
A detailed characterization of the actual seismic response of the prototype structure is then
offered using accurate numerical models. Refined finite element models are, in fact, realized
with the aid of SeismoStruct and SAP2000 computer codes to perform the fundamental
inelastic analyses: inelastic pushover analysis and inelastic dynamic time history analysis. An
excellent match between the DDBD hypotheses and the DTHA responses can be observed.
and important considerations regarding capacity design guidelines are pointed out.
Keywords: 3D seismic response; Displaced-Based Design; dual frame-wall structure; steel frame;
dynamic time-history;
Index
TABLE OF CONTENTS
Page
ABSTRACT ............................................................................................................................................i
TABLE OF CONTENTS .......................................................................................................................ii
LIST OF FIGURES ................................................................................................................................v
LIST OF TABLES..................................................................................................................................8
1 INTRODUCTION ...........................................................................................................................10
1.1 Research Objectives.................................................................................................................10
1.2 Presentation of the case study ..................................................................................................11
1.3 Organization of the report........................................................................................................13
2 DDBD DESIGN PROCEDURE .....................................................................................................14
2.1 Design Procedure Overview ....................................................................................................14
2.2 Elastic Response Spectra .........................................................................................................14
2.3 Design Preliminary Considerations .........................................................................................16
2.4 Transverse Direction Design....................................................................................................17
2.4.1 Step 1: Assignment of strength proportion between frames and walls and establishment
of wall inflection height. ..........................................................................................................17
2.4.2 Step 2: Determination of the displacement profile and equivalent SDOF system
characteristics...........................................................................................................................19
2.4.3 Step 3: Computation of the equivalent viscous damping and effective period ..............22
2.4.4 Step 4: Estimation of the design base shear and individual member strength ...............24
2.4.5 Step 5: Adoption of the capacity design provisions;......................................................27
2.5 Longitudinal Direction Design.................................................................................................31
2.5.1 Step 1: Assignment of strength proportion between frames and walls and establishment
of wall inflection height. ..........................................................................................................31
ii
Index
2.5.2 Step 2 and Step 3: Determination of the displacement profile and equivalent SDOF
system characteristics...............................................................................................................34
2.5.3 Step 4 and Step 5:Individual member strength and adoption of the capacity design
provisions to control higher mode effects................................................................................35
2.6 Closing remarks regarding DDBD procedure..........................................................................37
3 DESIGN OF PROTOTYPE STRUCTURE ....................................................................................38
3.1 Channel and Flanges Walls Design .........................................................................................38
3.2 Frame System Design ..............................................................................................................40
3.2.1 Preliminary Consideration on Seismic Design of Steel Members .................................41
3.2.2 Steel Column Design .....................................................................................................42
3.2.3 Steel Beam Design .........................................................................................................44
3.3 Design Considerations .............................................................................................................46
3.4 Closing remarks regarding the design of prototype structure ..................................................48
4 VERIFICATION OF NUMERICAL STRUCTURAL MODEL ....................................................49
4.1 SAP and SeismoStruct .............................................................................................................49
4.2 SeismoStruct models................................................................................................................51
4.2.1 Modelling consideration ................................................................................................51
4.3 SAP model ...............................................................................................................................58
4.4 Closing remarks regarding prototype structure’s numerical models .......................................60
4.5 Eigenvalue analysis..................................................................................................................60
4.5.1 Eigenvalue analysis in SAP ...........................................................................................60
4.5.2 Eigenvalue analysis in SeismoStruct .............................................................................61
4.5.3 Comparison between SeismoStruct and SAP ................................................................66
4.6 Closing remarks regarding the verification of numerical structural models............................67
4.7 Modal deformed shapes ...........................................................................................................68
5 DESIGN VERIFICATION THROUGH PUSHOVER AND NONLINEAR TIME HISTORY
ANALYSIS...........................................................................................................................................70
5.1 Pushover analysis.....................................................................................................................70
5.1.1 Horizontal lateral load pattern........................................................................................71
5.1.2 Static Pushover analysis in SeismoStruct ......................................................................73
5.2 Verification of the Displacement–Based Designed Structure through Pushover Analisis ......73
5.2.1 Closing remarks .............................................................................................................75
5.3 Dynamic Time History Analysis..............................................................................................75
5.3.1 Dynamic input................................................................................................................76
5.3.2 Inelastic dynamic time history in SeismoStruct.............................................................79
5.4 Verification of the Displacement–Based Designed Structure through DTHA ........................79
iii
Index
5.4.1 Transverse direction: Displacement Profiles. ................................................................79
5.4.2 Transverse direction: Maximum Storey Drift. ...............................................................82
5.4.3 Transverse direction: Wall shear forces.........................................................................84
5.4.4 Transverse direction: Wall moments. ............................................................................87
5.4.5 Transverse direction: Frame shear forces in outer columns...........................................89
5.4.6 Transverse direction: Frame moments in outer columns. ..............................................92
5.4.7 Transverse direction: frame shear force and moment in inner column..........................93
5.4.8 Longitudinal direction: Wall shear force and moments.................................................99
5.5 Closing remarks regarding the seismic design verification ...................................................102
6 SOURCES OF UNCERTAINTY ASSOCIATED WITH RESEARCH FINDINGS ...................104
6.1 ELASTIC VISCOUS DAMPING .........................................................................................104
6.2 ROLE OF FLOOR DIAPHRAGMS......................................................................................105
6.3 U-SECTION MODELLING UNCERTAINTIES .................................................................111
7 CONCLUSIONS ...........................................................................................................................115
7.1 Displacement–Based Designed method for dual frame-wall structure..................................115
7.2 Hints for new capacity design guidelines for frame-wall structure .......................................116
7.3 Future research.......................................................................................................................117
REFERENCES ...................................................................................................................................116
APPENDIX A.....................................................................................................................................118
APPENDIX B.....................................................................................................................................122
APPENDIX C.....................................................................................................................................124
iv
Index
LIST OF FIGURES
Page
Figure 1.1 Seismic behaviour for building structures...............................................................10
Figure 1.2 Plan view and mass distribution of prototype structure ..........................................11
Figure 1.3 Prototype’s structure structural layout.....................................................................12
Figure 2.1 Design Elastic Acceleration and Response Spectra ................................................15
Figure 2.2 Lateral force and moment distribution in a frame-wall dual system [Paulay, 2002]
...........................................................................................................................................16
Figure 2.3 Distribution of Shear Forces and Overturning Moment..........................................18
Figure 2.4 DDBD yield, plastic and total displacement profile................................................20
Figure 2.5 DDBD Inelastic Displacement Response Spectrum................................................23
Figure 2.6 Distribution of Shear forces and Overturning moment ...........................................26
Figure 2.7 Simplified Capacity Design Envelopes for Cantilever Walls .................................28
Figure 2.8 Dynamic Amplification Factor................................................................................31
Figure 2.9 Wall moment increment from link-beam action .....................................................33
Figure 2.10 Wall Moment Profiles of condensed wall element in the longitudinal direction ..33
Figure 2.11 DDBD yield, plastic and total displacement profile..............................................34
Figure 2.12 Simplified Capacity Design Envelopes for Cantilever Walls ...............................36
Figure 3.1 Moment–Curvature charts for 8m and 4m walls.....................................................39
Figure 3.2 Steel sections’class for flexural design....................................................................41
Figure 3.3 Frame column’s groups: Plan View. .......................................................................42
Figure 3.4 Geometric Steel sections parameter ........................................................................43
Figure 3.5 Inner columns’ new orientation...............................................................................44
Figure 3.6 Plan view of moment input for biaxial attack to two-way frame interior column ..47
Figure 3.7 Austrian cross-shape section ...................................................................................48
v
Index
Figure 4.1 Local chord system [SeismoStruct, 2007]...............................................................49
Figure 4.2 Fibre element model[SeismoStruct, 2007] ..............................................................50
Figure 4.3 Stress-Strain model for the structural materials adopted in SeismoStruct ..............51
Figure 4.4 Numerical model’ structural layout ........................................................................53
Figure 4.5 Model scheme used to represent U-shape wall system [Beyer et al., 2008] ...........53
Figure 4.6 General 3D view of SeismoStruc model .................................................................54
Figure 4.7 Rigid diaphragm constraints....................................................................................55
Figure 4.8 Link element independent degree of freedom .........................................................57
Figure 4.9 SeismoStruct models ...............................................................................................58
Figure 4.10 SAP model.............................................................................................................59
Figure 4.11 SeismoStruct Eigenvalue analysis scheme ............................................................62
Figure 4.12 Modal deformed shape: pure translational modes.................................................68
Figure 4.13 Modal deformed shapes: torsional modes .............................................................69
Figure 5.1 Capacity curve example ..........................................................................................71
Figure 5.2 Capacity curves obtained performing pushover analysis ........................................75
Figure 5.3 Design and artificial earthquakes’ Acceleration Response Spectra ........................76
Figure 5.4 Design and artificial earthquakes’ Displacement Response Spectra.......................76
Figure 5.5 Artificial record acceleration time-histories............................................................77
Figure 5.6 Fourier Amplitude Spectra ......................................................................................78
Figure 5.7 Displacement profile: comparison between DDBD hypothesis and DTHA
maximum response ...........................................................................................................80
Figure 5.8 Diplacement profile: comparison between DDBD hypothesis and DTHA avarege
response.............................................................................................................................81
Figure 5.9 Drift profiles: comparison between DDBD hypothesis and DTHA average
response.............................................................................................................................83
Figure 5.10 Wall shear profile: comparison between capacity envelope and DTHA maximum
response.............................................................................................................................85
Figure 5.11 Wall shear profiles: comparison between DDBD hypothesis and DTHA avarege
response.............................................................................................................................86
Figure 5.12 Wall moment profile: comparison between DDBD hypothesis and DTHA
maximum response ...........................................................................................................87
Figure 5.13 Wall moment profile: comparison between DDBD and DTHA avarage response
...........................................................................................................................................88
vi
Index
Figure 5.14 Frame shear profile: comparison between DDBD hypothesis and DTHA
maximum response ...........................................................................................................90
Figure 5.15 Frame shear profile: comparison between DDBD hypothesis and DTHA avarege
response.............................................................................................................................91
Figure 5.16 DTHA maximum moment curves for outer columns in transverse direction .......92
Figure 5.17 Comparison between DDBD, capacity design and DTHA response moment
profiles ..............................................................................................................................93
Figure 5.18 Frame shear profile: comparison between DDBD hypothesis and DTHA average
response.............................................................................................................................95
Figure 5.19 Frame moment profile: comparison between DDBD hypothesis and DTHA
average response ...............................................................................................................96
Figure 5.20 Wall and frame action during EQK_1 record at the time interval [8.20;8.80]......97
Figure 5.21 Wall and frame action during EQK_1 record at the time intervals [8.8 0;9.8] and
[9.8;9.98]...........................................................................................................................98
Figure 5.22 DTHA maximum shear experienced by 4m wall in longitudinal direction ........100
Figure 5.23 DTHA maximum moment experienced by 4m wall during in longitudinal
direction ..........................................................................................................................100
Figure 5.24 Average of maximum moment experienced by inner columns during DTHA ...101
Figure 5.25 Shear and moment capacity design envelopes proposed by Goodsir [1985] ......102
Figure 6.1 DTHA’s wall displacement profiles in transverse direction: comparison between
rigid and flexible diaphragms conditions........................................................................106
Figure 6.2 DTHA’s wall displacement profiles in transverse direction: comparison between
rigid and flexible diaphragms conditions........................................................................107
Figure 6.3 Displacement profiles assumed under flexible diaphragm conditions by wall and
inner pilastrades during DTHA in transverse direction. .................................................108
Figure 6.4 Displacement profiles assumed under flexible diaphragm conditions by wall and
inner pilastrades during DTHA in transverse direction. .................................................109
Figure 6.5 Different schemes for subdividing the U-shaped section into planar wall section
[Beyer et al., 2008]..........................................................................................................112
vii
Chapter 2. DDBD Design Procedure
LIST OF TABLES
Page
Table 2.1 Preliminary calculation to determine the contraflexure height HCF..........................19
Table 2.2 Design Displacement Information ............................................................................21
Table 2.3 Equivalent Viscous Damping Information ...............................................................23
Table 2.4 DDBD characterization of the equivalent SDOF structure ......................................24
Table 2.5 DDBD Shear Forces and Overturning Moment .......................................................25
Table 2.6 Moment and Shear capacity Envelopes ....................................................................29
Table 2.7 Moment and shear design actions for frame’s beam ................................................30
Table 2.8 Beams and Columns Final Design Action................................................................31
Table 2.9 Preliminary Calculation to determine contraflexure height HCF...............................32
Table 2.10 Equivalent SDOF Substiture Structure...................................................................34
Table 2.11 DDBD design action on wall and frame condensed structural elements................35
Table 2.12 Capacity Design action for wall in the longitudinal direction ................................35
Table 2.13 Capacity design action for external frames.............................................................36
Table 2.14 Capacity Design action for internal frames ............................................................37
Table 3.1 Shear and Moment capacities of 8 m and 4 m walls ................................................38
Table 3.2 Column group DDBD design actions .......................................................................42
Table 3.3 Selected shape profile for column sections...............................................................43
Table 3.4 Percentage difference between flexural strength demand and flexural strength
capacity .............................................................................................................................44
Table 3.5 DDBD design strength demand for beam.................................................................45
Table 3.6 Selected Shape profile for beam sections .................................................................45
Table 3.7 Percentage difference between flexural strength demand and flexural strength
capacity .............................................................................................................................45
8
Chapter 2. DDBD Design Procedure
Table 3.8 Gravity beam design action ......................................................................................46
Table 3.9 Selected Shape profile for gravity beam sections.....................................................46
Table 3.10 Percentage difference between flexural strength demand and flexural strength
capacity .............................................................................................................................46
Table 4.1 Beam and wall tributary masses ...............................................................................56
Table 4.2 Eigenvalue results for SAP model ............................................................................61
Table 4.3 Eigenvalue results for Link Element SeismoStruct model .......................................63
Table 4.4 Eigenvalue results for Link Element SeismoStruct model .......................................64
Table 4.5 Eigenvalue results for Equal DOF SeismoStruct model...........................................65
Table 4.6 Eigenvalue results for Equal DOF SeismoStruct model...........................................65
Table 4.7 Eigenvalues comparison between SAP and SeismoStruct .......................................66
Table 5.1 Spreadsheet sample of modal pattern distribution (longitudinal direction)..............72
Table 5.2 SeismoStruct link element model: pushover analysis results....................................74
Table 5.3 SeismoStruct equal DOF model: pushover analysis results.....................................74
Table 5.4 Displacement profiles: comparison between DDBD hypothesis and DTHA avarege
response.............................................................................................................................82
Table 5.5 Drift profiles: comparison between DDBD hypotheses and DTHA response .........84
Table 5.6 Wall moment profile: comparison between DDBD hypotheses and DTHA response
...........................................................................................................................................89
Table 6.1 Initial periods of 3D models with and without flexible diaphragms. .....................110
Table C.0.1 Wall shear forces: comparison between DDBD hypotheses and DTHA response
.........................................................................................................................................125
Table C.0.2 Frame shear forces: comparison between DDBD hypotheses and DTHA response
.........................................................................................................................................125
Table C.0.3 Frame moments: comparison between DDBD hypotheses and DTHA response
.........................................................................................................................................126
Table C.0.4 Frame shear forces: comparison between DDBD hypotheses and DTHA response
.........................................................................................................................................126
Table C.0.5 Frame moments: comparison between DDBD hypotheses and DTHA response
.........................................................................................................................................127
9
Chapter 2. DDBD Design Procedure
1 INTRODUCTION
1.1 Research Objectives
The principal aim of this research is to test and verify the recent direct displacement-based
design techniques specifically developed for wall-frame structures, complex but extremely
efficient lateral resistant system.
Engineering judgement in seismic regions indicates the dual frame-wall structures as an
excellent lateral force resisting system, able to guarantee an efficient control both on drift and
displacement deformations. Moreover, the use of interacting cantilever walls and frames
provides the retention of a satisfactory energy dissipation capacity during earthquake motions.
In addition, if it is extended over the full height of the building, the system can also boast a
great maintenance of lateral strength and stiffness properties.
Despite its advantages, the adoption of mixed systems is quite rare, and the use of separate
structural walls or frames tends to be preferred. The reasons of this apparent contradiction can
be found in the difficulties encountered to provide well established guidelines or design
provisions specifically dedicated to. In fact, the direct redaction of specific codes is thwarted
by the presence of complex interaction phenomena occurring between frame and wall during
shaking motions. Moreover, the dual systems are constituted as sum of two components
(frame and wall) characterized by very different response under lateral cyclic load, as Figure
1.1 testifies. As a consequence, the prediction of the actual interaction and global seismic
behaviour is extremely difficult and their use avoided.
a) Frame Building
b) Wall Building
Figure 1.1 Seismic behaviour for building structures
10
Chapter 2. DDBD Design Procedure
Experimental and numerical studies specifically focused on dual system structures are also
rare and scarce, usually referred to 2D systems homogeneously in RC material.
The study’s purpose is, then, to offer a concrete contribution for the comprehension of this
particular resistant system, analysing a three dimensional prototype structure. The attention
will be concentrated to a dual mixed frame-wall structure where two-way steel frames are
connected to channel walls of reinforced concrete. A well detailed description of the
prototype structure is offered in the next paragraph.
1.2 Presentation of the case study
The case study is defined as a twelve-storeys building, with an inter-storey height of 4 m for
the ground floor and 3.2 m at all other levels. As shown in Figure 1.2, the building plan is
organized around a regular bay module of 8 m x 8 m, for a global dimension of 32 m in xdirection and 24 m in y-direction.
500 t
8m
8m
8m
4m
11 @ 3.2 m = 35.2 m
10 @ 700 t
3 @ 8 m = 24 m
Flanges
Walls
Channel
Wall
4m
4 @ 8 m = 32 m
(a) Plan Dimensions
770 t
4m
8m
8m
8m
8m
(b) Masses and Heights
Figure 1.2 Plan view and mass distribution of prototype structure
The seismic structural system consists of two-way moment-resisting structural-steel frames
with channel walls of reinforced concrete at each end of the building containing elevators,
stairs and toilets. In particular, the U-shape core structures are defined by a channel wall of 8
m x 0.30 m and two wall flanges of 4 m x 0.30 m. Extending over the full building height, the
entire lateral resistant system clearly satisfies the code requirements of a regular configuration
both in plan and in elevation.
In the Figure 1.2 part (b) is schematized the mass distribution along the entire height of the
building. In particular, an estimation of the storey masses, including allowance for seismic
live loads and wall weight suggests to assign 770 tonnes at level 1, 700 tonnes at levels 2 to
11 and 500 tonnes at roof level.
Concerning the structural material adopted, the following mechanical properties will be
considered in the design process:
11
Chapter 2. DDBD Design Procedure
-
Concrete:
Reinforcing Steel:
Structural Steel:
f’c = 30 MPa;
fy = 400 MPa; fu = 1.35 fy; Es = 200GPa;
fy = 350 MPa; fu = 1.35 fy; Es = 200 GPa;
Anticipating one of the most important features of the design methodology adopted (DDBD
procedure) to guarantee the respect of strength hierarchy and to assure the complete
development of plastic-hinges zones according to the selected collapse mechanism, the
expected yield values are preferred instead of the nominal ones. Therefore, the following
values will be adopted in the design procedure:
-
Concrete:
Reinforcing Steel:
Structural Steel:
f’ce = 1.3 f’c = 39 MPa;
fye = 1.1 fy = 440 MPa;
fye = 1.1 fy = 385 MPa;
Figure 1.3 Prototype’s structure structural layout
Important preliminary observations can be offered observing the structural layout and
analysing separately the lateral resistant system characterizing the two principal directions:
the transverse and the longitudinal one. In the transverse (short) direction, there are three steel
frames acting in parallel to the end channel walls. While in the longitudinal direction can be,
instead, observed the presence of two external steel frames and two dual wall-framestructures. In this way the structural layout can be schematized as shown in Figure 1.3,
distinguishing the lateral resistant systems present in transverse and longitudinal directions.
Another relevant structural aspect regards the types of connection existing between the beam
and the wall systems. Gravity steel beams simply supported at both ends connect the channels
to the corner column, and thus do not induce seismic actions in the reinforced concrete walls.
In the longitudinal direction, the two internal steel frames are connected to the ends of the
wall flanges with steel beams that are moment-resisting at the columns, but pinned to the wall
flanges. As a consequence, although no moments will be transmitted to the wall by the frame
12
Chapter 2. DDBD Design Procedure
elements, the seismic shear acting on steel beams will induce moments at the channel axis,
reducing the base moment demand in the channel weak axis.
1.3 Organization of the report
Following the study’s aim to offer a concrete contribution for the comprehension of dualframe system, the research will face all the phases characterizing an usual design
methodology: the presentation of the structural layout in exam, the description of the
particular design procedure followed, the determination of the design actions, the design of
each seismic-resistant element, the definition of an appropriate numerical models and finally
the verification of the design hypothesis through non-linear static and dynamic analyses.
After the presentation of the case study structure proposed in Chapter 1, the design procedure
followed is considered in detail in Chapter 2. In particular, the DDBD design procedure is
illustrated with a particularly attention to the peculiar feature specifically developed for dual
frame-wall system. Dealing with a three-dimensional structure, the design procedure will be
then separately applied in both the principal directions: the transversal and the longitudinal
one.
In Chapter 3, an opportune combination of the orthogonal seismic actions guarantees a detail
design for all the structural elements: walls, columns and beams. The design procedure will be
essentially based on flexural strength requirements, even if some comments on shear strength
capacity are offered. Preliminary considerations complete and reinforce the design hypothesis.
Using SeismoStruct [v.4.0.9 built 992] and SAP [v. 10.0.1 Advantage] computer codes,
several finite element models of the prototype structure are realized. In particular, in Chapter
4, different peculiar feature are investigated and a sensitive analysis is presented. As results,
two different SeismoStruct models are selected as definitive models. The calibration and
validation of these models was performed in accordance to the results proposed by SAP2000
computer code.
Finally, a detailed characterization of the actual seismic response of the designed prototype
structure is offered. Following this purpose, two different types of analysis are exploited:
inelastic pushover analysis and inelastic dynamic time history analysis (IDTHA). A complete
response study based on a direct comparison between the expected behaviour and the actual
one is presented in Chapter 5. In this context, some limitations and inefficiency of current
capacity design guidelines are individuated and highlighted.
After the development of a complete study on three dimensional seismic response of a dual
frame-wall structure, Chapter 6 is dedicated to highlight and identify the issues which might
add uncertainty to the analysis outcomes. In fact, the verification procedures have a purely
analytical character and some approximations have been made during the design process and
the non-linear time-history analyses.
In the final chapter, the main findings of this report are summarised and the further research
needs are identified. Three Appendices are also included, presenting the most important
results not directly inserted in the previous Chapters.
13
Chapter 2. DDBD Design Procedure
2 DDBD DESIGN PROCEDURE
Considering singularly each principal directions, the entire DDBD design procedure is applied
to the prototype structure. Dealing with a three-dimensional structure, the design procedure
will be then separately applied in both the principal directions: the transversal and the
longitudinal one. The design in transverse direction will be, then, considered in detail; while,
as it is largely repetitive, only the most important phases and results obtained in the
longitudinal direction will be discussed.
2.1 Design Procedure Overview
Following the schematization suggested by Sullivan [2006], the displacement-based design
procedure for mixed building structure can be briefly summarized into 5 main steps:
-
STEP 1:
-
STEP 2:
-
STEP 3:
STEP 4:
STEP 5:
Assignment of strength proportion between frames and walls and
establishment of wall inflection height;
Determination of the displacement profile and equivalent SDOF system
characteristics;
Computation of the equivalent viscous damping and effective period;
Estimation of the design base shear and individual member strength;
Capacity design provisions;
The previous design procedure is applied separately in the two principal directions, taking
into consideration the layout features that singly characterized them. In fact, as mentioned in
section 1.2, the main difference can be traced in the type of interaction established between
frames and walls. In the transverse direction, frames and walls are basically present as parallel
working individual elements, while in the longitudinal direction two effective frame-wall
structures joint with link-beam are foreseen. Therefore, the general procedure for mixed
building system will be properly detailed and adapted with respect to the particular direction
analysed.
2.2 Elastic Response Spectra
It was assumed the building is to be constructed on a soft-soil layer in a moderate seismicity
region with PGA = 0.35 g. The elastic acceleration and displacement response spectra
adopted are described in equations from (2.1) to (2.5) and respectively depict in Figure 2.1.
a. Elastic acceleration response spectrum:
0 ≤ T ≤ TA
T A ≤ T ≤ TB
⎡
T ⎤
S A (T ) = PGA ⋅ ⎢1 + (C A − 1) ⋅ ⎥
TA ⎦
⎣
S A (T ) = C A ⋅ PGA
(2.1)
(2.2)
14
Chapter 2. DDBD Design Procedure
TB
T
T ⋅T
S A (T ) = C A ⋅ PGA ⋅ B 2 C
T
S A (T ) = C A ⋅ PGA ⋅
TB ≤ T ≤ TC
T ≥ TC
where:
S A (T )
PGA
CA
and
(2.3)
(2.4)
is the spectral acceleration expressed in units of ‘g’;
is the peak ground acceleration, in this case sets equal to 0.35 g;
is the multiplier factor for PGA to obtain peak response acceleration, CA 2.5;
TA,TB and TC are set respectively equal to 0.25 sec, 1.0 sec and 5.0 sec.
1.2
1.4
1.0
1.2
Displacement S D (T) [ m ]
Acceleration S A (T) [ g ]
b. Elastic displacement response spectrum:
The elastic displacement response spectrum is obtained in accordance to the following
equation:
T2
S D (T ) =
⋅ S A (T ) ⋅ g
(2.5)
4π 2
0.8
ξ = 0.05
0.6
0.4
0.2
0.0
1.0
ξ = 0.05
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
5
6
7
0
Period T [ sec ]
(a) Elastic Acceleration Response Spectrum
1
2
3
4
5
6
7
Period T [ sec ]
(b) Elastic Displacement Response Spectrum
Figure 2.1 Design Elastic Acceleration and Response Spectra
Should be noticed that the corner period TC is assumed to be 5.0 sec in obedience to the
more up-to-date information provided in recent work by Faccioli et al. [2004]. In fact,
using selected sets of high-quality digital strong motion data from different world regions
(Taiwan, Japan, Italy, and Greece), has been highlighted how the salient features of
displacement response spectra in the long-period range (up to 10 s period) are essentially
function of magnitude, source distance, and site conditions. In particular, the corner period
appears to increase linearly with magnitude with conservative values offers by the
following relationship:
Tc = 1.0 + 2.5( M w − 5.7)
[seconds]
(2.6)
valid for earthquakes with moment magnitude greater than Mw=5.7. Therefore, a corner
period set equal to 5.0 sec seems to better fit the recent observations, considering a
moderate-large earthquake events in the magnitude range 5.4<Mw<7.6 . However, will be
found that this assumption does not influence in any way the design as the effective
structural period is sensibly lower than this value.
15
Chapter 2. DDBD Design Procedure
2.3 Design Preliminary Considerations
Promoting a specific plastic collapse mechanism (cinematically admissible), the aim of
DDBD method is to guarantee a high performance of the structure under earthquake attack,
limiting the displacement and drift deformation experienced. More in general, the purpose of
this methodology is to control the level of damage sustained by the system, with respect to the
selected limit state. In this study, the design process will be ruled by the damage-control limit
state characterized by a design drift limit equal to θC = 0.02, as many national codes
suggested.
An attractive design solution could result in the adoption of the same dimensions for all the
beams at all levels except the roof, where a reduction of 50% in the beam strength is
appraised. The reasons of this design choice rely on the identical seismic strength demands
which the beam would experience at the ultimate limit state.
The use of the same beam dimensions implies that all the columns, although carrying different
axial loads, would be subjected to near identical moment demand [Paulay, 2002]. Therefore,
the same nominal storey moment capacities are requested to each frame and the
corresponding nominal shear forces will be constant up to the height of the building. The
lateral force and moment distribution suggested and adopted in the design strategy are, then,
illustrated in the Figure 2.2, where the contribution of walls and frames systems to the total
lateral resistance strength are singularly depicted.
Figure 2.2 Lateral force and moment distribution in a frame-wall dual system [Paulay, 2002]
The DDBD focused on dual wall-frame structure foresees the knowledge of the beam depth hb
already from the firsts steps of the procedure. Some are, in fact, the initial equations where
this parameter is explicitly required (i.e. the yield drift of steel frame). But, since an accurate
estimation of hb is completely premature at this stage, an iterative procedure was carried out
using successive estimation of the section depth. An iterative procedure is, therefore,
performed until convergence is achieved. In the next pages, the results of the last iteration are
presented, where the beam depth hb equal to 0.65 m is obtained.
For completeness sake should be mentioned that alternative processes has been proposed for
W-series ASCE steel group section [Sullivan et al., 2002] in order to avoid preliminary
iterative procedures, but no relationship is available for the European steel group
classification adopted in this project.
16
Chapter 2. DDBD Design Procedure
2.4 Transverse Direction Design
The transverse design is now considered in detail. For modelling purposes, at the initial
stages, the presence of three distinct steel frame and two separate RC channel walls will be
replaced by only two condensed elements, mentioned as condensed steel frame and condensed
RC wall or just with Frame and Wall labels, for simplicity sake.
2.4.1
Step 1: Assignment of strength proportion between frames and walls and
establishment of wall inflection height.
As stated in section 2.3, beams of constant strength are assigned up the height of the structure,
except for the top where beam strengths are set equal to be 50% of those of the lower levels.
Therefore the frame storey shear will be constant along the entire the height of the building
and the internal column will carry twice the moment and the shear of the external ones.
Moreover, since the stiffness of walls above the base plastic hinges guarantees adequate
protection against such a soft-storey mechanism, the base columns’ strength are assigned to
provide an inflection height of 0.5 the storey height, as in the upper levels.
The last strength assignment regards the lateral load repartition between the walls and frames
system. Considering the number of the frames and the dimensions of the channel walls, it was
established to allocate 40% of the total base shear to the frames and the remaining 60% to the
walls system. Henceforward, the proportion factor βF is set equal to 0.4 and the following
equivalences subsist:
V Frame = β F ⋅ Vbase = 0.4 ⋅ Vbase
and
VWall = (1 − β F ) ⋅ Vbase = 0.6 ⋅ Vbase
(2.7)
Expressed as function of unit base shear, the initial stages of analysis are summarized in Table
2.1. The lateral force, shear force and overturning moment are listed both with respect to the
entire structure and to the Frame and Wall condensed elements.
As initial hypothesis, the displacement vector is assumed varying linearly with the height.
Therefore, indicating with mi the storey mass and with Hi the respective height at each level,
the lateral force will be consequently proportional to miHi/ΣmiHi, as listed in Col. 5.
In the next columns (Col.6 and Col.7), the total shear force and the total overturning moment
are respectively shown. In particular, the total shear forces VTi are found summing the relative
forces above the level considered, while the overturning moments MOTM,i are evaluated as :
M OTM ,i = ∑ F j ⋅ (H j − H i )
12
(2.8)
j =1
Recalling the selected proportion factor βF, the Frame VF,i and Wall shear forces VW,i are
evaluated at each level adopting equation 2.7. Finally, the wall moment profile is calculating
using the following equation:
M w,i = M w,i + 1 + Vw,i +1 ⋅ (H i +1 − H i )
(2.9)
Analysing the vertical moment profile (see Figure 2.3), it is possible to predict the exact
position of the contraflexure point. Characterized by a null value of the moment and by a
change in the sign of the profile distribution, this point can be found interpolating linearly the
data referred to level 6 and 7, as suggest by Col.11 in Table 2.1. Hence:
H CF = 22.0m
(2.10)
17
Chapter 2. DDBD Design Procedure
40
40
40
38
38
38
36
36
36
34
34
34
32
32
32
30
30
30
28
28
28
26
26
26
Heigth [ m ]
24
22
20
18
Heigth [ m ]
42
24
22
20
18
24
22
20
18
16
16
16
14
14
14
12
12
12
10
10
8
8
6
6
4
4
4
2
2
2
0
0.00
0
0.00
10
Frame
Shear
Force
8
6
Wall
Shear
Force
0.25
0.50
0.75
1.00
0.25
0.50
0.75
Shear Force
Shear Force
Total Overturning Moment
Frame Overturning Moment
0
-0.50
1.00
42
42
40
40
40
38
38
38
36
36
36
34
34
34
32
32
32
30
30
30
28
28
28
26
26
26
24
24
24
20
18
22
20
18
16
14
14
12
12
12
10
10
10
8
8
8
6
6
6
4
4
2
2
2
0
0.0
0
0.0
5.0
10.0
15.0
20.0
Moment
25.0
30.0
5.0
10.0
15.0
20.0
0.75
1.00
18
14
Wall
OTM
0.50
20
16
Frame
OTM
0.25
22
16
4
0.00
Wall Overturning Moment
42
22
-0.25
Shear Force
Heigth [ m ]
Heigth [ m ]
Wall Shear Force
Frame Shear Force
42
Heigth [ m ]
Heigth [ m ]
Total Shear Force
42
25.0
30.0
0
-5.0
HCF
0.0
5.0
Moment
10.0
15.0 20.0 25.0 30.0
Moment
Figure 2.3 Distribution of Shear Forces and Overturning Moment
18
Chapter 2. DDBD Design Procedure
Table 2.1 Preliminary calculation to determine the contraflexure height HCF
1
2
3
4
5
6
Total
shear
force
7
Total
overturning
moment
8
9
10
11
Level
Height
Mass
miHi
Lateral
force
Frame
shear
Frame
Moment
Wall
shear
Wall
moment
Hi
mi
miHi
Fi
VTi
MOTM,i
VF,i
MF,i
VW,i
MW,i
[ - ]
[ m ]
[ t ]
[ tm ]
12
39.2
500
19600
[ rel ]
0.1127
[ rel ]
[ rel ]
[ - ]
[ - ]
[ - ]
[ - ]
0.1127
0
0.4
0.000
-0.287
0.00
11
36.0
700
25200
10
32.8
700
22960
0.1449
0.1320
0.2576
0.36
0.4
1.280
-0.142
-0.92
0.3897
1.19
0.4
2.560
-0.010
-1.37
9
29.6
700
8
26.4
700
20720
18480
0.1192
0.5089
2.43
0.4
3.840
0.109
-1.41
0.1063
0.6151
4.06
0.4
5.120
0.215
-1.06
7
23.2
6
20.0
700
700
16240
0.0934
0.7085
6.03
0.4
6.400
0.309
-0.37
14000
0.0805
0.7890
8.30
0.4
7.680
0.389
0.62
5
4
16.8
13.6
700
11760
0.0676
0.8567
10.82
0.4
8.960
0.457
1.86
700
9520
0.0548
0.9114
13.56
0.4
10.240
0.511
3.32
3
2
10.4
700
7280
0.0419
0.9533
16.48
0.4
11.520
0.553
4.96
7.2
700
5040
0.0290
0.9823
19.53
0.4
12.800
0.582
6.73
1
4.0
770
3080
0.0177
1.0000
22.67
0.4
14.080
0.600
8.59
0
0
0
0
0.0000
1.0000
26.67
-------- ------------- ------------- ------------- ------------- ------------- ---------------Sum
8270
173880
1
0.4
15.680
0.600
10.99
------------- ------------- ------------- ------------------
2.4.2
Step 2: Determination of the displacement profile and equivalent SDOF system
characteristics
The yield curvature of the walls φy,Wall can be obtained using the following expression:
φ y ,Wall = K1
εy
[ m-1]
lWall
(2.11)
where εy the yield strain for reinforcing steel, lWall the wall length and K1 a proportionality
factor depending of the cross section of the wall and the longitudinal reinforcement layout.
Although several researchers have derived expression for the K1-factor from momentcurvature analysis for a number of different wall geometries and reinforcement layout, the
yield curvature of the walls φy,Wall is calculated using the original expression introduced by
Priestley [2003] valid for flanged wall, for I-section walls and for T-section walls when flange
is in compression:
φ y ,Wall = K1
ε y 1.5 ⋅ ε y
=
= 0.000413 [ m-1]
lWall
lWall
(2.12)
However, even if not available at the time in which this study was performed, more up-to-date
expressions of the yield curvature have been recently proposed by Beyer et al. [2008]
conducting specific laboratory tests on U-shape RC walls.
Using the wall inflection height HCF, the displacement profile at yield can be established in
accordance to the following equations:
19
Chapter 2. DDBD Design Procedure
⎛H2
H i3 ⎞
⎟
Δ yi = φ y ,Wall ⋅ ⎜⎜ i −
6 ⋅ H CF ⎟⎠
⎝ 2
2
⎛ H i ⋅ H CF H CF
Δ yi = φ y ,Wall ⋅ ⎜⎜
−
2
6
⎝
For H i ≤ H CF
For H i ≥ H CF
(2.13)
⎞
⎟
⎟
⎠
(2.14)
The actual yield displacement profile is indicated in Col.4 of Table 2.2, where all the
numerical results regarding design displacements are collected and presented.
Once the yield displacement profile is known, the field of plastic deformation has to be
explored to determine the final design displacement profile. For this reason, the design drift
that accounts for higher modes effects has to be estimated. More in detail, the design drift
θd,lim should be selected as the minimum between the code drift limit θC and that associated to
the maximum capacity curvature of the walls θWall. In particular, θWall can be approximated to
the wall drift limit at contraflexure height θCF, evaluated as:
θ CF =
φ y ,Wall ⋅ H CF
+ (φ dc − φ y ,Wall ) ⋅ L p = 0.0244
2
(2.15)
Where:
φy,Wall is the wall limit- state curvature;
LP
is the plastic hinge length at the base of the wall;
is the strength penetration length;
LSP
k
is a reduction factor;
Detailed information regarding the above parameters are available in APPENDIX A, where a
complete description will comment each term.
12
11
10
9
8
Total Displacement
Level
7
6
Yield Displacement
5
Plastic Displacement
4
3
2
1
0
0.00
0.40
0.80
Displacement [ m ]
Figure 2.4 DDBD yield, plastic and total displacement profile
Since θCF exceed the code drift limit of (θC=0.02), the code drift limits govern the wall design.
Nevertheless, this value should be properly reduced considering the relevant height of the
20
Chapter 2. DDBD Design Procedure
building. An opportune drift correction factor ωθ will be applied and the design drift limit
becomes equal to:
⎡ ⎛ n − 5 ⎞ ⎛ M OTM , F
⎞⎤
θ d = θ d ,lim ⋅ ωθ = θ d ,lim ⋅ ⎢1 − ⎜
+ 0.25 ⎟⎟⎥ = 0.01882
⎟ ⋅ ⎜⎜
⎢⎣ ⎝ 100 ⎠ ⎝ M OTM ,Total
⎠⎥⎦
(2.16)
Finally, the design displacement profile is given by the equation 2.16 and listed in Col.5 of
.
φ y ,Wall ⋅ H CF
⎛
Δ Di = Δ y ,i + ⎜⎜θ d −
2
⎝
⎞
⎟⎟ ⋅ H i
⎠
(2.17)
In the Figure 2.4, the yield, plastic and total DDBD displacement profile are depicted. Should
be noticed how the plastic displacement amount, defined as the second addend of equation
2.16, represents the most relevant contribution to the total displacement profile.
Table 2.2 Design Displacement Information
1
2
3
4
5
Level
Height
Mass
Yield
Displacement
Design Displacement
Profile
Hi
mi
Δyi
ΔDi
mi Δ2Di
[ m ]
[ t ]
[m]
[m ]
[tm ]
0.145
0.130
0.116
0.101
0.087
0.072
0.057
0.043
0.030
0.019
0.010
0.003
0.000
0.705
0.645
0.584
0.524
0.464
0.404
0.343
0.283
0.225
0.167
0.112
0.060
0.000
[ - ]
12
11
10
9
8
7
6
5
4
3
2
1
0
39.2
500
36.0
700
32.8
700
29.6
700
26.4
700
23.2
700
20.0
700
16.8
700
13.6
700
10.4
700
7.2
700
4.0
770
0.0
0
--------------- -----------------------------------Sum
8270
------------------
6
------------------
0.000
2
7
8
mi ΔDi
mi ΔDi Hi
[tm]
[tm ]
248.3
352.4
290.8
451.2
239.0
409.0
192.2
366.8
150.6
324.6
114.0
282.5
82.5
240.3
56.2
198.4
35.3
157.2
19.6
117.2
8.8
78.7
2.8
46.4
0.0
0.0
-------------- -------------------1440.13
3024.67
2
13813.12
16241.51
13414.57
10857.52
8570.38
6553.15
4806.16
3333.55
2138.48
1218.73
566.56
185.60
0.00
----------------81699.35
With the displacement profile at maximum response established and the floor weights and
heights known, a complete characterization of the equivalent SDOF substitute structure can
be performed defining the design displacement Δd, the effective mass me and effective height
he as shown in Eq. (2.18) to Eq. (2.20).
∑ (m
12
Design
Dispiacement
Design Dispiacement:
Δd =
j =1
i
12
∑ (m
j =1
i
⋅ Δ2i
)
⋅ Δi )
= 0.476m
(2.18)
21
Chapter 2. DDBD Design Procedure
12
me =
Effective mass:
∑ (m
j =1
i
⋅ Δi )
= 6353tonnes
Δd
(2.19)
12
He =
Effective height:
∑ (m Δ h )
i
j =1
i i
12
∑ (m Δ )
j =1
i
= 27.0m
(2.20)
i
2.4.3 Step 3: Computation of the equivalent viscous damping and effective period
Using a secant stiffness representation of structural of response, the displacement-based
seismic design requires a modification to the elastic displacement response spectrum to
account for ductile response. The influence of the ductility can be represented by a viscous
damping ξsys related to the overall response of the entire system. This important parameter can
be, then, expressed as weighted mean of the equivalent viscous dampings referred to the
Frame ξFrame and Wall ξWall condensed elements, as shown in equation 2.21.
ξ sys =
ξWall M OTM ,Wall + ξ Frame M OTM , Frame
M OTM ,total
(2.21)
With ξFrame and ξWall calculated as:
⎛ μWall − 1 ⎞
⎛μ
−1⎞
⎟⎟
and
(2.22)
ξ Frame = 0.05 + 0.577⎜⎜ Frame ⎟⎟
⎝ μWall π ⎠
⎝ μ Frameπ ⎠
Where the terms μWall and μFrame indicate respectively the ductility demands on Wall and
Frame. The equations 2.23 to 2.26 address the methodology adopted to evaluate μWall and
μFrame.
Δd
μWall =
(2.23)
Δ y ,Wall
ξ Wall = 0.05 + 0.444⎜⎜
μ Frame =
(θ
Δd
He
y , Frame
)
(2.24)
The symbols Δ y,Wall and θ y, Frame point the yield displacement of SDOF system and the steel
frame yield drift, defined in accordance the following relationships:
⎛H H
H2 ⎞
Δ y ,Wall = φ y ,Wall ⎜⎜ CF e − CF ⎟⎟
6 ⎠
⎝ 2
0.65ε y Lb
θ y , Frame =
hb
With Lb and hb respectively the beam’s length span and the beam’s section depth.
(2.25)
(2.26)
Listed with respect to the Wall and Frame condensed elements, the yield displacement, the
ductility demand and the equivalent viscous damping are respectively shown in Table 2.3.
22
Chapter 2. DDBD Design Procedure
Also it is declared the final value assumed by the system equivalent viscous damping ξsys,
obtained as result of equation 2.20 previously mentioned.
Table 2.3 Equivalent Viscous Damping Information
Data Description
a) Wall:
Wall Yield displacement
Wall Ductility Demand
Wall_Eq. Viscous Damping
b) Frame:
Frame Yield Drift
Frame Ductility Demand
Frame Eq. Viscous Damping
c) System
Eq. System Viscous Damping
Symbol
Value
Unit
Δy,Wall
μWall
ξWall
0.089
5.333
16.48
[m]
[ - ]
[%]
θyFrame
μFrame
ξFrame
0.015
1.145
7.32
[ - ]
[ - ]
[%]
ξsys
11.10
[%]
Taking into account the hysteretic properties of the system, the damping modifier Rξ is
adopted in order to reduce the Elastic Displacement Response Spectrum. Function of the
equivalent viscous damping, the modification factor Rξ is expressed in accordance to the
previous revision (1998) of Eurocode EC8:
⎛ 0.07
Rξ = ⎜
⎜ 0.02 + ξ
sys
⎝
⎞
⎟
⎟
⎠
0.5
= 0.731
(2.27)
Displacement Response Spectrum
Displacement Spectrum SD (T) [ m ]
1.4
ξ =5.0%
1.2
1.0
ξ = 11.1%
0.8
0.6
ΔD
0.4
0.2
0.0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Te
Period T [ sec ]
Figure 2.5 DDBD Inelastic Displacement Response Spectrum
In fact, although a more recent expression is offered in the 2003 revision of Eurocode EC8 for
the damping modifier Rξ, a recent analysis conducted by Priestley et al. [2007] supports the
1998 EC8 expression for artificial spectral compatible earthquake records and actual
23
Chapter 2. DDBD Design Procedure
accelerograms without near-field forward directivity velocity pulse characteristics. Therefore,
since Eq. (2.27) does provide the best representation of the accelerograms used in this study,
it is adopted here only in order to obtain the most valid verification of the design procedure.
Finally with respect to the design displacement Δd, the effective period Te of the SDOF
substitute structure is directly estimated from the reduced displacement Response Spectrum as
shown in Errore. L'origine riferimento non è stata trovata..
2.4.4 Step 4: Estimation of the design base shear and individual member strength
With the effective period established, the effective stiffness Ke and design base shear Vbase are
calculated using the following equations 2.27 and 2.28:
4π 2 me
Te2
= Ke × Δ D
Ke =
(2.28)
Vbase
(2.29)
Now, the characterization of the SDOF substituting structure is complete and Table 2.4
summarizes the results. Observing the data proposed in that table, should be noticed how the
effective mass me represents the 77% of the building entire mass estimated as 8270 tonne (see
Table 2.1), while the base shear Vbase represents the 16.40% of the total weight.
With the effective value of the design base shear known, the total shear forces and
overturning moment acting at each level of the structure can be easily evaluated considering
also the results obtained in the previous stages (see Table 2.1). For clearness sake, Table 2.5
will summarize the final results using the same scheme already adopted: firstly are shown the
data related to the entire structure and then that referred to the condensed Frame end Wall
element.
The results are also graphically expressed in Figure 2.6 following the criteria illustrated in
Figure 2.3.
Table 2.4 DDBD characterization of the equivalent SDOF structure
Data Description
Symbol
Value
Unit
Design Displacement
ΔD
0.48
[m]
Effective height
He
27.01
[m]
Eq. System Ductility
μsys
3.66
[ - ]
Eq. System Viscous Damping
ξsys
0.11
[ - ]
Effective Period
Te
3.00
[ sec ]
Effective Mass
me
6353
[ tonnes ]
Effective Stiffness
Ke
27.93
[ MN/m ]
Vbase
13.30
[ MN ]
Base Shear
Now, abandoning the condensed Frame and Wall elements, the estimation of the design
actions will be referred to the actual seismic resistant system effectively constituted by walls,
24
Chapter 2. DDBD Design Procedure
columns and beams. Consequently, the following four sections are request to completely
define the flexural and shear design of each singular structural member.
(a) Wall Based Flexural design
From theErrore. L'origine riferimento non è stata trovata. Table 2.1, the total wall-base
moment is:
M Wall , Base = 10.99 × Vbase = 146.2 MNm
(2.30)
This is shared between the two channel walls, resulting in a design moment of 73.1 MN m
per wall.
Table 2.5 DDBD Shear Forces and Overturning Moment
1
2
6
7
Total
overturning
moment
8
9
10
11
Level
Height
Total shear
force
Frame shear
Frame
Moment
Wall shear
Wall
moment
Hi
VTi
MOTM,i
VF,i
MF,i
VW,i
MW,i
[ - ]
[ m ]
[ MN ]
[ MN m ]
[ MN ]
[ MN m ]
[ MN ]
[ MN m ]
12
11
10
9
8
7
6
5
4
3
2
1
0
39.2
36.0
32.8
29.6
26.4
23.2
20.0
16.8
13.6
10.4
7.2
4.0
0
1.50
3.43
5.18
6.77
8.18
9.42
10.49
11.39
12.12
12.68
13.06
13.30
13.30
0.0
4.8
15.8
32.3
54.0
80.2
110.3
143.9
180.4
219.2
259.8
301.6
354.8
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
0.00
17.02
34.05
51.07
68.10
85.12
102.15
119.17
136.20
153.22
170.24
187.27
208.55
-3.8
-1.9
-0.1
1.4
2.9
4.1
5.2
6.1
6.8
7.4
7.7
8.0
8.0
0.0
-12.2
-18.3
-18.7
-14.1
-4.9
8.2
24.8
44.2
66.0
89.5
114.3
146.2
(b) Wall Based Shear design
Recalling the proportion factor βF, the expected shear sustained by the walls system is
equal to
VWall ,base = (1 − β F )Vbase = 8.0MN
(2.31)
Also in this case shared between the two walls, individual base shear of 4 MN per wall is
expected.
(c) Frame Beam Flexural Design
The total shear force carried by the frames is calculated as equation 2.7 suggests:
V Frame,base = β F Vbase = 5.3MN
(2.32)
The shear force for each frame is hence equal to 1.77 MN. Considering the design
assumption of equal beam strength up to the building (excepted for the top level) and the
presence of 6 potential plastic hinges at each level per frame, the beam flexural design
will state a beam flexural strength equal to:
for storey beam
M bi , storey =
VF H i
= 946 KNm
6
(2.33)
25
Chapter 2. DDBD Design Procedure
M bi ,roof = 0.5 × M bi , storey = 473KNm
for roof beam
Total Shear Force
Wall Shear Force
42.0
40.0
40.0
38.0
38.0
38.0
36.0
36.0
36.0
34.0
34.0
34.0
32.0
32.0
30.0
30.0
28.0
28.0
28.0
26.0
26.0
26.0
Total
Shear
Force
Heigth [ m ]
24.0
22.0
20.0
18.0
Heigth [ m ]
42.0
40.0
30.0
Heigth [ m ]
Frame Shear Force
42.0
32.0
24.0
22.0
20.0
18.0
24.0
22.0
20.0
18.0
16.0
16.0
14.0
14.0
12.0
12.0
12.0
10.0
10.0
10.0
8.0
8.0
8.0
6.0
6.0
6.0
4.0
4.0
4.0
2.0
2.0
2.0
0.0
0.00
0.0
0.00
16.0
14.0
Wall
Shear
Force
Frame
Shear
Force
5.00
10.00
15.00
5.00
10.00
0.0
-5.00
15.00
Total Overturning Moment
0.00
5.00
10.00
Shear Force [ MN ]
Shear Force [ MN ]
Shear Force [ MN ]
Total Overturning Moment
Wall Overturning Moment
42.0
42.0
42.0
40.0
40.0
40.0
38.0
38.0
38.0
36.0
36.0
36.0
34.0
34.0
34.0
32.0
32.0
32.0
30.0
30.0
30.0
28.0
28.0
28.0
26.0
26.0
24.0
22.0
20.0
18.0
24.0
Heigth [ m ]
Heigth [ m ]
Total
OTM
26.0
Heigth [ m ]
(2.34)
22.0
20.0
18.0
24.0
22.0
20.0
18.0
16.0
16.0
16.0
14.0
14.0
14.0
12.0
12.0
12.0
10.0
10.0
10.0
8.0
8.0
8.0
6.0
6.0
6.0
4.0
4.0
4.0
2.0
2.0
Frame
OTM
2.0
Wall
OTM
0.0
0.0
0.0
100.0
200.0
300.0
Moment [ MN m ]
400.0
0.0
100.0
200.0
300.0
400.0
0.0
-200.0
Moment [ MN m ]
HCF
0.0
200.0
Moment [ MN m ]
Figure 2.6 Distribution of Shear forces and Overturning moment
26
400.0
Chapter 2. DDBD Design Procedure
(d) Frame Column Flexural Design
Assuming that the beam moments are equally distributed above and below the beamcolumn joints, the strength capacity of interior columns will result two times higher than
external columns. Moreover, in order to maintain constant the shear profile at each level
(also at the taller ground storey), the moment capacity of the columns at the base will need
to be calculated as:
M C ,base = Vcol H 01 − 0.5∑ M bi
(2.35)
Hence:
for outer columns
M C ,base= 709 KNm
for inner columns
M C ,base= 1419 KNm
For the outer columns, the following equation is used:
M C, f =
where:
½
2
MBj
1
2 M Bj
2
(2.36)
is the factor that take into account the equal repartition of the induced beam
moments above and below the joint connection;
is the amplification factor for take into account biaxial attack (borrowed from
DDBD guidelines focused on RC frame);
is the beam moments;
With regards to the inner column, values equal to two times the results previously
obtained will be adopted for the design purposes.
2.4.5 Step 5: Adoption of the capacity design provisions;
Strictly, the capacity design provisions should not be carried out until the design requirements
in the orthogonal direction are defined. However, for simplicity sake, the capacity provisions
will be applied separately in both the principal direction in order to obtain design solicitations
reliable enough to allow the propose of an immediate design solution. For this reason,
abandoning the condensed elements simplification, all the results presented in the next
sections will be directly expressed in function of the effective structural elements: channel
walls and steel frames.
(a) Capacity Design for Walls
Starting from the moment capacity design, a bilinear envelope is adopted. Three the
crucial points to consider in the profile: the base, the mid height and the top. In particular,
the overstrength base moment capacity φ0MB is assigned at the base level, the overstrength
moment M0Wall,0,5H defined the mid-height point and zero moment is assigned at the top of
the wall. So defined, the moment capacity profile is illustrated in Figure 2.7 part (a).
Moreover, taking into consideration the possible presence of inclined flexure/shear
diagonal tension stress, a tension shift envelope is considered, moving upwards (i.e.
27
Chapter 2. DDBD Design Procedure
“shifting”) the entire moment profile for a quantity equal to lW/2, where lW is the length of
the wall.
Shear Force Capacity Envelope
DDBD Moment Profile
Capacity Envelope
Tension Shift
Overstrenght Moment Capacity
DDBD Shear Force
Shear Capacity Envelope
40.0
40.0
38.0
38.0
36.0
36.0
34.0
34.0
32.0
32.0
30.0
30.0
28.0
28.0
26.0
26.0
24.0
24.0
Height [ m ]
Heigth [ m ]
Moment Capacity Envelope
22.0
20.0
18.0
M°0.5H
22.0
20.0
18.0
16.0
16.0
14.0
14.0
12.0
12.0
10.0
10.0
8.0
8.0
6.0
6.0
4.0
4.0
2.0
2.0
0.0
-20.0
0.0
20.0
40.0
60.0
80.0
100.0
120.0
Overstrength Shear Force
0.0
-6.0
-1.0
φ° M W, b
Flexural Moment [ MN m ]
4.0
9.0
V° base =°>°° ° V V bas e
14.0
19.0
Shear Force [ MN]
(a) Moment Capacity Envelope
(b) Shear Force Capacity Envelope
Figure 2.7 Simplified Capacity Design Envelopes for Cantilever Walls
As mentioned before, since zero moment is assigned at the top, the knowledge of only two
points is necessary to plot out the capacity envelope: the overstrength moment at the base
level φ0MB and the overstrength moment at the mid-heights M0Wall,0,5H. In order to
evaluate these moments the following equations are adopted:
0
0
M Wall
,base = φ ⋅ M Wall ,base
(2.37)
0
0
M Wall
, 0.5 H = C1,T ⋅ φ ⋅ M B
(2.38)
Where:
φ°
is the overstrength factor equal to 1.2;
C1,T is a dynamic amplification factor expressed as function of fundamental elastic
period Ti=Te/(μsys)0.5:
C1,T = 0.4 + 0.075Ti ⋅ (μ sys − 1)
(2.39)
More simple is the trace of the shear force capacity envelope, where a straight line defines
directly the entire profile, joining the capacity-design shear force at base level , V°base , to
the design-shear force at the top of the wall, V°top. The following equations from 2.39 to
2.43 are adopted in this case:
0
0
VWall
(2.40)
,base = φ ωV VWall ,base
0
0
VWall
,top = C 3VWall ,base
(2.41)
28
Chapter 2. DDBD Design Procedure
where ωV and C3 are dynamic amplification factor respectively equal to:
ωV = 1 +
μ
C 2,T
φ0
(2.42)
C 2,T = 0.067 + 0.04(Ti − 0.5) ≤ 1.15
with
(2.43)
C 3 = 0.9 − 0.3Ti ≤ 1.15
(2.44)
The final moment and shear design capacity envelopes are summarized in the Table 2.6,
expressed, as usual, as function of the level storey height.
Table 2.6 Moment and Shear capacity Envelopes
Wall moment
Overstrength
Moment
Capacity
Capacity
Evnvelope
Moment
Tension Shift
Wall shear
Overstrength
Shear Force
MW,i
φ MW,i
CE_MW,i
TS_MW,i
VW,i
φ VW,i
[ - ]
[ MN m ]
[ MN m ]
[ MN m ]
[ MN m ]
[ MN ]
[ MN ]
[ MN ]
12
11
10
9
8
7
6
5
4
3
2
1
0
0.00
-6.11
-9.14
-9.36
-7.05
-2.47
4.10
12.38
22.10
32.98
44.75
57.15
73.11
0.00
-7.34
-10.97
-11.23
-8.45
-2.96
4.92
14.85
26.52
39.58
53.71
68.57
87.73
0.00
8.50
17.00
25.50
34.01
42.51
51.01
57.17
62.99
68.81
74.63
80.45
87.73
10.63
19.13
27.63
36.13
44.63
52.80
58.62
64.44
70.26
76.08
81.91
87.73
87.73
-1.91
-0.95
-0.07
0.72
1.43
2.05
2.59
3.04
3.40
3.68
3.87
3.99
3.99
-2.29
-1.14
-0.08
0.87
1.72
2.46
3.10
3.64
4.08
4.42
4.65
4.79
4.79
5.50
6.17
6.84
7.51
8.19
8.86
9.53
10.21
10.88
11.55
12.23
12.90
13.74
Level
0
0
Capacity
Envelope
CE_V
0
W
(b) Capacity Design for Frames
With the beam flexural strength established (step 3), the shear solicitation acting on beam
elements can be calculated using equation 2.44:
Vb , i =
2 ⋅M b.i
(l B − hc )
(2.45)
Where the symbols lB and hc refer to the beam length spam and the column section depth.
Since at this stage the complete knowledge of beam and column geometric section
properties is not possible, all the actions will be referred to the element barycentre line.
Therefore, the previous expression becomes:
Vb , i =
2 ⋅M b.i
lB
Taking into consideration also the contribution due to vertical load, the shear capacity
provisions foresee a design shear diagram varying linearly with span length (see equation
2.44). In order to maximize that expression, a value x equal to zero has to be selected,
corresponding to the beam section at column centreline.
29
Chapter 2. DDBD Design Procedure
Vb ( x) =
where:
φ0
0
w
G
2φ 0 M b ,i
lB
+
wG0 l B
− wG0 x
2
(2.46)
is the overstrength factor assumed equal to 1.2;
is the vertical load amplified of 30% for dynamic considerations;
Table 2.7 Moment and shear design actions for frame’s beam
Height
Lateral force
Storey
Frame shear
Storey
Frame OTM
Beam Flexural
Design
Storey Shear
Hi
Fi
Vs,i
MF,i
Mbeam,1
Vbeam,1
V
[ - ]
[ m ]
[ MN ]
[ MN ]
[ MN/m ]
[ KN m ]
[ KN ]
[ KN ]
12
11
10
9
8
7
6
5
4
3
2
1
0
39.2
36.0
32.8
29.6
26.4
23.2
20.0
16.8
13.6
10.4
7.2
4.0
0
1.50
1.93
1.76
1.58
1.41
1.24
1.07
0.90
0.73
0.56
0.39
0.24
0.00
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
0.00
0.00
5.67
11.35
17.02
22.70
28.37
34.05
39.72
45.40
51.07
56.75
62.42
69.52
472.9
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
0.0
118.2
236.5
236.5
236.5
236.5
236.5
236.5
236.5
236.5
236.5
236.5
236.5
0.000
274.7
416.5
416.5
416.5
416.5
416.5
416.5
416.5
416.5
416.5
416.5
416.5
0.0
---------------
------------------
-------------------
------------------
------------------
----------------------
--------------------
-------------------
13.30
1.77
69.52
Level
Sum
Beam Shear
Design
0
B,max
2719
Table 2.7 summarizes the results obtained. Should be noticed how the design assumption of
equal steel beam size at all levels (hence equal strength) simplifies considerably the design
process.
The required column flexural and shear strength to satisfy capacity design requirements may
be taken as:
φ f M c ≥ M 0 = ω f φ f0 M CE
(2.47)
φV VC ≥ V 0 = ω f φV0VCE
(2.48)
with:
MCE the corresponding column moment resulting from the design frame shear force;
VCE the shear corresponding to the design frame shear force;
φ0f and φ0V
overstrength factor for flexure and shear design set equal to 1.1 and 1.2;
ωf
is the dynamic amplification factor.
The dynamic amplification factor is considered uniformly equal to 1.3 at all the levels
except at the base level where a value equal to 1 is adopted, as shown in Figure 2.8.
Should be underlined that the above provisions may not provide an absolute security against
the column hinging at the levels above the base. However, this eventuality will prove as not
critical since the stiffness of walls, which remain essentially elastic above the base hinge,
will protect the building against soft storey mechanisms.
Considering that a quite uniform design moment profile is expected, the same column size
is adopted up the height of the building. In fact, the base column has a moment demand that
is already 51% higher than at other levels (to provide the required shear force in the ground
30
Chapter 2. DDBD Design Procedure
floor) and in the upper levels the design forces are amplified by a capacity design factor
equal to 1.3xφ0 = 1.56.
40.0
35.0
Heigth [ m ]
30.0
25.0
20.0
15.0
10.0
5.0
First Storey
0.0
0.00
0.50
1.00
1.50
2.00
Dynamic Amplification Factor
Figure 2.8 Dynamic Amplification Factor
For simplicity sake, the complete schemes of beam and column design action are listed in
APPENDIX A, while the beams and columns final design actions are summarized in Table
2.8.
Shear
Table 2.8 Beams and Columns Final Design Action
VC1,des
461.08
Outer Column
COLUMN
Moment
Shear
BEAM
Moment
[ KN ]
Inner Column
VC2,des
922.2
[ KN ]
Outer Column
MC1,des
1043.30
[ KN m ]
Inner Column
MC2,des
2086.61
[ KN m ]
Storey beam
V0B,max,s
416.5
[ KN ]
Roof Beam
V0B,max,r
274.7
[ KN ]
Storey beam
Mbeam,s
945.8
[ KN m ]
Roof Beam
Mbeam,r
472.9
[ KN m ]
2.5 Longitudinal Direction Design
Since it is largely repetitive, the whole detailed design will not be presented here; however,
the most important stages of the steps will be treat and the main results comment.
2.5.1
Step 1: Assignment of strength proportion between frames and walls and
establishment of wall inflection height.
Allowing efficiency in design and construction, the same beam size are selected at all the
levels (except the roof), guaranteeing the same beam flexural strength along the entire height
of the building. Therefore, designed for the same strength capacity, a total number of 28 beam
plastic hinge locations at each floor can be considered in the longitudinal direction. Moreover,
a minor wall moment capacity is expected due to the reduced wall section dimensions in
31
Chapter 2. DDBD Design Procedure
longitudinal direction (4m instead of 8m). These two considerations indicate that a higher
percentage of base shear should be allocated to the frames in this direction. Hence a
proportion factor βF equal to 0.5 is adopted.
Calculations similar to those performed in section 2.4.1, allow the entire compilation of Table
2.9, except for the Col. 12 where the effects of link-beam moment are taking into account.
Table 2.9 Preliminary Calculation to determine contraflexure height HCF
1
2
3
4
5
6
Total
shear
force
7
Total
overturning
moment
8
9
10
12
Frame
shear
Frame
Moment
Wall
shear
Wall
moment
MF,i
VW,i
MW,i
Height
Mass
miHi
Lateral
force
Hi
mi
miHi
Fi
VTi
MOTM,i
VF,i
[ - ]
[ m ]
[ t ]
[ tm ]
[ rel ]
[ rel ]
[ rel ]
[ - ]
[ - ]
[ - ]
[ - ]
12
11
10
9
8
7
6
5
4
3
2
1
0
39.2
36.0
32.8
29.6
26.4
23.2
20.0
16.8
13.6
10.4
7.2
4.0
0
500
700
700
700
700
700
700
700
700
700
700
770
0
19600
25200
22960
20720
18480
16240
14000
11760
9520
7280
5040
3080
0
0.113
0.145
0.132
0.119
0.106
0.093
0.081
0.068
0.055
0.042
0.029
0.018
0.000
0.113
0.258
0.390
0.509
0.615
0.709
0.789
0.857
0.911
0.953
0.982
1.000
1.000
0.000
0.361
1.185
2.432
4.061
6.029
8.296
10.821
13.563
16.479
19.530
22.673
26.673
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.000
1.600
3.200
4.800
6.400
8.000
9.600
11.200
12.800
14.400
16.000
17.600
19.600
-0.387
-0.242
-0.110
0.009
0.115
0.209
0.289
0.357
0.411
0.453
0.482
0.500
0.500
-0.114
-1.354
-2.243
-2.711
-2.797
-2.542
-1.989
-1.179
-0.152
1.051
2.387
3.816
5.702
Level
In fact, as mentioned in section 2.3 and depicted in Figure 2.9, the seismic shear in the pinned
link-beam induces secondary moments in the walls that can be evaluated at the centreline axis
as:
( M bl − M br )lW ,CL
M b , wall = M bl +
(2.49)
Lb
Where:
lW,CL
is the distance from the integral column centreline to the wall axis;
Lb
is the span bay length;
Mbl and Mbr are the moment at the right and left end of the link-beam, that for equal
positive and negative moment capacities in the beam can be measured as:
VF H S
(2.50)
nbe
The global effect of these link-beam induced moments can be traced as a reduction of the
moment demand in the lower regions of the wall but as an increment in the upper regions.
Consequently, a lowering in the elevation of the contraflexure height is expected, as shown in
Figure 2.10, where are compared the moment profiles considering or not the presence of
beam-link.
Mb =
Regarding the contraflexure point, Figure 2.9 shows how a lowering of almost one level (3m)
has to be taking into account. In particular, the exact position of contraflexure height can be
determined applying the same procedure indicated in section 2.4.1 but considering the
modified moment profile due to the presence of link-beam induced moment (dark blue line in
Figure 2.9). A visual analysis suggests immediately the contraflexure point’s location as very
32
Chapter 2. DDBD Design Procedure
close to the fourth floor level, therefore the expected contraflexure height HCF will
approximately result around at 13 m.
Figure 2.9 Wall moment increment from link-beam action
Clearly appears, by now, the importance of the layout configuration in the DDBD procedure.
In the transverse direction, where frames and walls work in parallel without any direct link
and the proportion factor βF is set equal to 0.4, the Wall base moment is evaluated equal to
10.99 unit of base shear x m (see Table 2.1) and the contraflexure point is situated at 22.00 m.
On the other hand, in longitudinal direction where the presence of link-beam is beheld and βF
increases to 0.5, a drastically reduction is observed both in MW,base and in HCF. The wall basemoment decreases to 5.72 unit of base shear x m and the contraflexure point is around 13 m.
33
Chapter 2. DDBD Design Procedure
12
No link beam
11
With link beam
10
9
8
Level
7
6
5
4
3
2
HCF
1
0
-5.00
-3.00
-1.00
1.00
3.00
5.00
7.00
Wall Moment
Figure 2.10 Wall Moment Profiles of condensed wall element in the longitudinal direction
2.5.2 Step 2 and Step 3: Determination of the displacement profile and equivalent SDOF
system characteristics.
With the contraflexure height exactly established (HCF = 12.95 m), the procedure continues
following the indications already adopted in transverse direction.
The detailed table of design displacement’s data is provided in APPENDIX A, while the final
DDBD design displacement profiles are shown in Figure 2.11.
Table 2.10 Equivalent SDOF Substiture Structure
Parameter Description
Symbol
Value
Design Displacement
ΔD
0.48
[
m
]
Effective height
He
26.97
[
m
]
Eq. System Ductility
μsys
2.56
[
-
]
Eq. System Viscous Damping
ξsys
0.09
[
-
]
Effective Period
Te
2.79
[ sec
Effective Mass
me
6372
[ tonnes ]
Effective Stiffness
Ke
32.36
[ MN/m ]
Vbase
15.51
[
Base Shear
Unit
MN
]
]
Avoiding tedious repetitions, the most important parameters for the characterization of the
equivalent SDOF substitute structure, such as the design displacement ΔD, the equivalent
viscous damping ξsys,the effective period Te, the effective stiffness Ke and finally the base
shear Vbase are summarized in the Table 2.10.
34
Chapter 2. DDBD Design Procedure
12
11
10
9
Level
8
7
Total Displacement
6
Yield Displacement
5
Plastic Displacement
4
3
2
1
0
0.000
0.500
1.000
Displacement [ m ]
Figure 2.11 DDBD yield, plastic and total displacement profile
2.5.3
Step 4 and Step 5:Individual member strength and adoption of the capacity design
provisions to control higher mode effects.
Sharing the DDBD design shear forces between the walls and the frames in accordance to the
repartition factor βF, the following actions are attributed to each condensed structural
member:
Table 2.11 DDBD design action on wall and frame condensed structural elements
1
2
3
Height
Total shear
force
Hi
VTi
[ - ]
[ m ]
12
11
10
9
8
7
6
5
4
3
2
1
0
39.2
36.0
32.8
29.6
26.4
23.2
20.0
16.8
13.6
10.4
7.2
4.0
0
Level
4
Total
overturning
moment
5
6
7
8
Frame shear
Frame
Moment
Wall shear
Wall moment
MOTM,i
VF,i
MF,i
VW,i
MW,i
[ MN ]
[ MN m ]
[ MN ]
[ MN m ]
[ MN ]
[ MN m ]
1.748
3.997
6.046
7.895
9.543
10.992
12.242
13.291
14.140
14.790
15.239
15.514
15.514
0
5.596
18.387
37.734
62.997
93.536
128.712
167.885
210.416
255.665
302.992
351.759
413.816
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
0.000
24.823
49.646
74.469
99.292
124.115
148.938
173.761
198.584
223.407
248.230
273.053
304.081
-6.0
-3.8
-1.7
0.1
1.8
3.2
4.5
5.5
6.4
7.0
7.5
7.8
7.8
0.0
-19.2
-31.3
-36.7
-36.3
-30.6
-20.2
-5.9
11.8
32.3
54.8
78.7
109.7
Moving to a complete definition of the design actions, the capacity design provisions are now
applied following the procedure illustrated in section 2.4.5.
35
Chapter 2. DDBD Design Procedure
Now, while the capacity design for frame does not present any particularity and the results are
directly given in Appendix A, an important observation should be advanced for the wall
capacity design profiles. Considering the capacity envelope in part (a) of Figure 2.12, results
clear the mentioned procedure does not provide enough safeguard in the estimation of the
design moment envelope. In fact, based only on the estimation of the base and mid-height
moments, the bilinear profile is not able to cover the DDBD moment diagram in the upper
regions of the wall. Therefore, in this case, the DDBD and Overstrength moment profile
appear more conservative than the capacity envelope proposed, as highlighted by the solid
and dashed grey line in the following chart (Figure 2.12).
However, since the objective of this study is to evaluate the trustworthiness of DDBD
methodology, the solicitations proposed by this type of capacity design procedure will be
adopted for the design of the prototype structure (see Table 2.12).
Table 2.12 Capacity Design action for wall in the longitudinal direction
Shear
base level
V0base
5543.88
[ KN ]
Moment
base level
M0base
26537.37
[ KN m ]
4 m WALL
Moment Capacity Envelope
Shear Force Capacity Envelope
Capacity Envelope
DDBD Shear Force
Overstrenght Moment Capacity
Tension Shift
Shear Capacity Envelope
40.0
40.0
38.0
38.0
36.0
36.0
34.0
34.0
32.0
32.0
30.0
30.0
28.0
28.0
26.0
26.0
24.0
24.0
H eigh t [ m ]
Heigth [ m ]
DDBD Moment Profile
22.0
20.0
18.0
M 0.5H
16.0
-40.00
-30.00
-20.00
-10.00
22.0
20.0
18.0
16.0
14.0
14.0
12.0
12.0
10.0
10.0
8.0
8.0
6.0
6.0
4.0
4.0
2.0
2.0
0.0
0.00
Overstrength Shear Force
10.00
20.00
φ M w ,base
Flexural Moment [ MN m ]
30.00
40.00
-4.00
-2.00
0.0
0.00
2.00
4.00
6.00
8.00
V°base=φ° ωV Vbase
Shear Force [ MN]
(a) Moment Capacity Envelope
(b) Shear Force Capacity Envelope
Figure 2.12 Simplified Capacity Design Envelopes for Cantilever Walls
36
Chapter 2. DDBD Design Procedure
For completeness sake, also the final design actions for the frame system are summarized in
Table 2.13 and Table 2.14. Noticed that frame design procedure was properly detail and
adapted for the individual external frames and for the internal ones connected to the wall
system.
Table 2.13 Capacity design action for external frames
Shear
COLUMN
Moment
Shear
BEAM
Moment
Outer Column
VC1,des
432.2
[ KN ]
Inner Column
VC2,des
864.4
[ KN ]
Outer Column
MC1,des
977.92
[ KN m ]
Inner Column
MC2,des
1955.85
[ KN m ]
Storey beam
V0B,max
368.2
[ KN ]
Roof Beam
V0B,max
199.4
[ KN ]
Storey beam
Mbeam,1
886.5
[ KN m ]
Roof Beam
Mbeam,1
443.3
[ KN m ]
It should be recalled that, at this stage, all the actions are referred to the joint centroid
sections. Only, in a more advanced phase, with all the element’s dimensions known, will be
possible to determine the exact design solicitation acting at the column face. In fact, in order
to determine the actions at the real element’s ends, it will be sufficient to reduce the values
already found in proportion to the ratio of column width to beam span.
Table 2.14 Capacity Design action for internal frames
Shear
COLUMN
Moment
Shear
BEAM
Moment
Outer Column
VC1,des
995.7
[ KN ]
Inner Column
VC2,des
995.7
[ KN ]
Outer Column
MC1,des
1329.80
[ KN m ]
Inner Column
MC2,des
1329.80
[ KN m ]
Storey beam
V0B,max
470.5
[ KN ]
Roof Beam
V0B,max
265.8
[ KN ]
Storey beam
Mbeam,1
886.5
[ KN m ]
Roof Beam
Mbeam,1
443.3
[ KN m ]
2.6 Closing remarks regarding DDBD procedure
The DDBD method was applied to the case study structure considering separately the
transverse and the longitudinal directions. Due to the remarkable differences in the layout
features, the general procedure was properly detailed in function of the principal direction
considered. Particular importance was, hence, dedicated to the characterization of the two
orthogonal directions, distinguishing the number and the dimensions of the structural
elements and the type of interaction existing between them.
Finally, following the step indicated by Sullivan [2006], all the design solicitations are
completely defined.
37
Chapter 3. Design of Prototype Structure
3 DESIGN OF PROTOTYPE STRUCTURE
An opportune combination of the orthogonal seismic actions guarantees a detail design for all
the structural elements: walls, columns and beams. Due to the elevated number of elements to
be design, the chapter will be divided into two main sections singularly dedicated to the wall
and frame system design.
The design procedure will be essentially based on the satisfaction of flexural strength
requirements, even if some comments on shear strength member capacity are offered.
Preliminary verifications complete and reinforce the design hypothesis performed.
3.1 Channel and Flanges Walls Design
Although core structure are often used in reinforced concrete buildings as members providing
lateral strength and stiffness, experimental and numerical studies on their inelastic behaviour
under earthquake loading are scarce [Beyer et al., 2008]. Moreover, the DDBD procedure
followed is calibrated on common dual frame-wall structure, usually constituted by a single
panel wall and a parallel one-way frames. Therefore, instead of incurring in unfruitful
complication in the design process, the U-shape wall system configuration will be ideally split
into three separate members individually acting on their own plane parallel to the momentresisting frames linked to. Although appears quite rough, this simplification allows an
immediate initial sizing of the reinforcement rebar and a preliminary verification of the
geometric plan dimensions. Only in a second phase, with the aid of inelastic pushover
analyses and inelastic dynamic time-history analyses outcomes, will be possible to refined the
design procedure considering the core structure in its peculiar three-dimensional
configuration.
Therefore, according to this hypothesis the wall system design requires the definition of two
distinct cantilever walls’ typologies: the 8 m and the 4 m wall respectively in transverse and
longitudinal direction.
Table 3.1 Shear and Moment capacities of 8 m and 4 m walls
Required Strenght
Symbol
8 m WALL
4 m WALL
Unit
0
base
13739
5544
[ KN ]
0
base
87727
26537
[ KN m ]
Shear
V
Moment
M
Weight
N Wall Weigth
6477
1619
[ KN ]
Additive seismic axial
force
T
[-]
2549
[ KN ]
0
base
Design Axial Force
N
6477
4168
[ KN ]
Design curvature
φMAX
0.008
0.022
[m ]
-1
38
Chapter 3. Design of Prototype Structure
The flexural reinforcement design is computed considering the moment-axial load couples
acting at the base of each structural element. Concerning the axial action, should be noticed
that the vertical loads have to be distributed among all the vertical resistant elements and
therefore among both the walls and the columns. Therefore, assuming a uniform distribution
of the floor masses on the storey areas, the axial loads pertinent to the wall system are
estimated at the base level equal to 6477 kN for the 8m walls and 1619 kN for the 4m walls.
An additional axial force has, moreover, to be considered for the longitudinal wall design due
to link-beam interaction (see Figure 2.9). In fact, the beams develop a shear which is
transferred into the wall as a compressive (or tensile) force. Since the maximum value of the
shear is limited by the flexural strength and the beam’s length, an upper limit to the
compression (or tension) force imposed by the beams onto the wall could be obtained by
summing the maximum beam shears over the height of the wall.
0
base
M
Moment Curvature
Diagram
8m WALL
φMax,Des
M0base,Des
[-]
[ kN ]
0.008
88790
φMax,U
M0base,U
[-]
[ kN ]
0.0084
88920
φ max
(a) Moment-Curvature Chart for 8 m wall
Moment Curvature
Diagram
0
base
M
4m WALL
φMax,Des
M0base,Des
[-]
[ kN ]
0.022
25513
φMax,U
M0base,U
[-]
[ kN ]
0.024
27060
φ max
(b) Moment-Curvature Chart for 4 m wall
Figure 3.1 Moment–Curvature charts for 8m and 4m walls
The most severe design conditions for the base level impose the adoption of the maximum
compression load for the design process. Therefore, the design axial load N0base is obtained
39
Chapter 3. Design of Prototype Structure
summing the global effect T of the induced link-beam’s shear with the axial force associated
to the usual gravity load condition NWall, Weight, as shown in Table 3.1.
The reinforcement areas of each wall are obtained by the axial load-moment interaction
curves provided by the URC_RC [URC_RC, version1.0.2] program.
Especially for the transverse 8 m wall, the design is governed by the design maximum
curvature φMAX strongly influencing both the amount and spacing of transverse and
longitudinal reinforcement, as can be observed in the design details proposed in the following
sections:
(a) 8 m WALL
Considering a design axial load and a design base moment respectively equal to
N0base=6480 kN and M0base=87730 kN m, the reinforcement area is estimated as 122φ18
bars with a 134 mm spacing. In order to guarantee the design curvature requirement, a
small enlargement of the width length is necessary, moving from 0.30 m to 0.35 m. With
the new dimensions of the wall section 8 m x 0.35 m, the reinforcement ratio is equal to
1.11%, amply contained in the usual code range limit ( 0.3% < ρ < 2.0%).
(b) 4 m WALL
Assuming as the design action N0base=4170 kN and M0base=26540 kN m, the corresponding
reinforcement areas is estimated equal to 76φ18 bar with a 108 mm spacing. For
consistency purposes, also in this case the wall width is increased to 0.35 m, even if is not
strictly required by the design. As in the previous case the usual code range limits are
respected, providing a reinforcement ratio equal to 1.38%.
Even if quite consistent, the design reinforcement is just sufficient to guarantee the DDBD
moment-curvature requirements, as can be observed in Figure 3.1. In fact, the design
maximum curvature φMAX is reached in an advanced plastic phase, not excessively far from
the ultimate conditions. Although a higher ductility threshold should be preferred in common
conditions, the maintenance of the initial structural configuration imposes an arrangement
between the actual wall’s flexural capacity and the reinforcement’s geometric limitations.
As conclusion, should be emphasised that the moment-curvature analyses conducted and
performed on a component basis (i.e. looking at flange and web sections separately) does not
represent a definitive statement for the seismic design of the core structures. The aim of this
simplified M-curvature analyses is just to allow an initial sizing of the reinforcement rebar
and to verify the plan geometric configuration with a special attention to the wall width. Only
in a second phase, with the aid of inelastic pushover analyses and inelastic dynamic timehistory analyses, will be possible to check the design curvatures and the final reinforcement
quantities observing the actual seismic response of the U-shaped walls.
3.2 Frame System Design
A brief introduction on steel member seismic design will lead in this paragraph entirely
dedicated to the steel frame system design.
For clearness sakes the design procedure will be then illustrated in to two main sections: the
column design and the beam design.
40
Chapter 3. Design of Prototype Structure
3.2.1 Preliminary Consideration on Seismic Design of Steel Members
For steel elements is well-known the dependence of flexural capacity with respect to their
own geometric characterization. For this reason, in accordance to EC3 code, four element
classes are instituted in order to collect all the possible typologies of transverse sections. The
classification is performed on the base of the slenderness parameter λ , defined as:
λ=
b
t
fy
(3.1)
E
where b and t are respectively the width and the thickness of the compressed transverse
section elements. In particular, the local slenderness ratio b/t indicates the element sensibility
to the failure induced by local instability: for low values of the parameter, the section is
perfectly able to develop the full plastic condition while for high values local instability
occurs before the ultimate moment is reached, provoking brittle failures.
Therefore the steel members classification is performed subdividing in appropriate intervals
the range of all the possible values suitable for the slenderness parameter λ . As anticipating,
four distinct section classes are defined: the ductile sections (known also as Class1), the
compact sections (Class 2), the semi-compact sections (Class 3) and the slender sections
(Class 4). How shown in Figure 3.2, different section modulus have to be selected in the
design procedure for each element class. In particular, for Class 1 and Class 2 elements the
plastic modulus Wpl (Z in ASCE code) can be used, for Class 3 the elastic modulus Wel (S in
ASCE code) and only the effective one Weff should be considered for Class 4.
Hence, strictly speaking in term of flexural strength capacity, member of Class 1 and Class 2
are equivalent; their difference relies in the rotational capacity exploitable in plastic field: an
amply ranged is in fact allowed for Class 1 while a limited range is admitted for Class 2 (see
Figure 3.2 part b).
Following the prescriptions and the suggestions present in Eurocode and ASCE code
regarding the design of steel building in seismic zones, all the structural element designed will
be belong to Class 1.Therefore, two are the immediate consequences for the design procedure:
1)
The geometry of transverse sections will be governed by the slenderness limits
imposed for the λ parameter.
2)
The plastic modulus Wpl (Z in ASCE code) can be adopted in the design
calculations.
(a) Flexural strength as function of slenderness
(b) Moment-Curvature Diagrams
Figure 3.2 Steel sections’class for flexural design
41
Chapter 3. Design of Prototype Structure
3.2.2
Steel Column Design
Figure 3.3 Frame column’s groups: Plan View.
Combining the final DDBD actions in the two orthogonal directions, the layout of the steel
frame columns can be organized into three groups: the corner column, the lateral column and
the core column, as shown in Figure 3.3.
In accordance with those three groups, the design actions expected for each column are
summarized in the Table 3.2, where the symbols MCD,X and MCD,Y indicate respectively the
flexural moment required in the longitudinal direction (around y-axis)and in the transverse
direction (around x-axis).
Table 3.2 Column group DDBD design actions
Column Group
[-]
CORNER COLUMNS
LATERAL COLUMNS
CORE COLUMNS
Direction
MCD,i
MCD,X
MCD,Y
[ kN m ]
[ kN m ]
977.9
-
1955.8
1043.3
1329.8
2086.6
[-]
[ kN m ]
Transverse
-
Longitudinal
977.9
Transverse
1043.3
Longitudinal
1955.8
Transverse
2086.6
Longitudinal
1329.8
In Table 2.1, the presence of an empty cell captures the attention and deserves same
comments. In transverse direction, the original layout of the structure foresees pinned
connections at both the ends of the gravity beams between the corner columns and the channel
walls. Therefore no moment will be transmitted to the column at the beam ends. For this
reason, the DDBD design procedure can not explicitly provide any information regarding the
corner column’s flexural strength in this direction. Therefore, the relative cell in Table 3.3
will remain empty testifying again the peculiar features of case study structure strictly
correlated to the DDBD design actions. Obviously, an adequate flexural strength will be,
however, provided to the corner columns in transverse direction in accordance to common
engineering considerations.
42
Chapter 3. Design of Prototype Structure
Recalling the research’ scope to verify the consistency of direct displacement-based design
methodology, the design is successful in so far as the effective strength capacity of each
structural element has been closed to the design indications. Therefore, a detailed design of
steel members is then carry on until a close convergence between the strength demand and the
strength capacity is matched. Starting from the data available for HD shape profile, three not
standardize shape profile are defined and the full geometric characterization is listed in Table
3.3.
Should be noticed how the proposed I-shape profiles respond essentially only to the
requirements indicated by the DDBD design actions. No axial-moment interaction checks
have been considered, in fact, at this stage. Even if this simplification should appear quite
roughly, it is just related to the germ of the effective design procedure. After the detailed
pushover checks should be possible, in fact, to complete and improved the preliminary design
verifying both the axial-moment and torsional-moment interactions’ checks.
Table 3.3 Selected shape profile for column sections
Column group
h
bf
tw
tf
r
[-]
[ mm ]
[ mm ]
[ mm ]
Corner Column
350.0
300
15
22
27
Lateral Column
390.0
390
24
36
Core Column
400
400
25
43
A
IX
2
[ mm ] [ mm ] [ cm ]
Wel,x Wpl,x
4
3
3
IY
Wel,y Wpl,y
4
3
3
[ cm ]
[ cm ] [ cm ] [ cm ] [ cm ] [ cm ]
184.2
40491
2314
2608
9922
661
1016
27
363.4
96172
4932
5673
35650
1828
2795
15
424.4
117042
5852
6786
45913
2296
3492
Figure 3.4 Geometric Steel sections parameter
Despite the careful design, some percentage differences can be however observed between the
effective flexural strength capacities and the flexural strength demands, as shown in Table
3.4. The origin of these discrepancies can be ascribed to two main causes: the design adoption
of H shape section (characterized by deeply different properties with respect to the two
principal directions) and the respect of geometric ratio limits imposed for Class 1 steel section
element.
43
Chapter 3. Design of Prototype Structure
Table 3.4 Percentage difference between flexural strength demand and flexural strength capacity
DDBD Flexural Strength Demand
Design Flexural Strength Capacity
Percentage difference
Column Group
MCD,X
MCD,Y
MCD,X
MCD,Y
MCD,X
MCD,Y
[-]
[ KN m ]
[ KN m ]
[ KN m ]
[ KN m ]
[%]
[%]
Corner Column
977.9
N.A.
1004.0
391.0
2.67
N.A.
Lateral Column
1955.8
1043.3
2184.0
1076.0
11.66
3.14
Core Column
1329.8
2086.6
1344.5
2612.7
1.10
25.21
Notice that for the core columns the strength demand in transverse direction and MCD,Y is
grater than that in longitudinal direction MCD,X ( see Table 3.2). For this reason and
considering the peculiar geometric characteristic of I shape profile (i.e., I X>>IY), a change in
the orientation of the element section is established with respect to the original layout
configuration.
Figure 3.5 Inner columns’ new orientation
Therefore, rotating of 90° the element section, the local section strong axis X will lies on
global transverse direction, while the local weak axis Y will assume the global longitudinal
direction, as shown in Figure 3.5.
3.2.3 Steel Beam Design
Recalling the flexural and shear strength requirements indicated by the DDBD procedure, the
Table 3.5 summarized the design actions for beams both for transverse and longitudinal
direction.
On the base on some preliminary consideration about lateral-torsional buckling and on the
limited difference that separate the corresponding solicitations in the two orthogonal
44
Chapter 3. Design of Prototype Structure
directions, the same section’ size is selected for the storey and roof beams both in transverse
and longitudinal directions. This adoption will allow efficiency in the design process and in
the construction phase, even if is paid with a quite high percentage difference between the
effective strength demand and strength capacity in longitudinal direction.
Table 3.5 DDBD design strength demand for beam
Direction
Strength Demand
Beam Type
Symbol
[ KN ]
0
B,max
416.5
V
0
B,max
274.7
[ KN ]
Storey beam
Mbeam,1
945.8
[ KN m ]
Roof Beam
Mbeam,1
472.9
[ KN m ]
V
0
B,max
470.5
[ KN ]
V
0
B,max
265.8
[ KN ]
Storey beam
Mbeam,1
886.5
[ KN m ]
Roof Beam
Mbeam,1
443.3
[ KN m ]
Roof Beam
Moment Strength
Storey beam
Shear Strength
Roof Beam
Longitudinal
Direction
Unit
V
Storey beam
Shear Strength
Transverse
Direction
Value
Moment Strength
Guaranteeing the respect of geometric limitation dictated by the Class 1 requirements, the
following not standardize steel sections are individualized respectively for the storey and roof
beams.
Table 3.6 Selected Shape profile for beam sections
Beam group
[-]
h
bf
tw
tf
r
A
[ mm ] [ mm ] [ mm ] [ mm ] [ mm ] [ cm2 ]
IX
Wel,x
Wpl,x
IY
Wel,y
Wpl,y
[ cm4 ]
[ cm3 ]
[ cm3 ]
[ cm4 ]
[ cm3 ]
[ cm3 ]
Storey Beam
650
180
12
15
24
133.3
82868.0
2549.8
3018.3
1474.3
163.8
271.2
Roof Beam
400
180
8.6
13.5
21
84.5
23128.4
1156.4
1307.1
1317.8
146.4
229.2
Table 3.7 Percentage difference between flexural strength demand and flexural strength capacity
Demand
Capacity
Percentage difference
Direction
Beam type
MCD,X
MCD,X
MCD,X
[-]
[-]
[ KN m ]
[ KN m ]
[%]
Storey Beam
946
1162
22.9
Roof Beam
473
503
6.4
Storey Beam
887
1162
31.1
Roof Beam
443
503
13.5
Transverse Direction
Longitudinal Direction
The last elements to be designed are the gravity beams jointing the 8m wall with the corner
column in transverse direction (Figure 3.5). Pinned connected at both the ends, these elements
not participate to the seismic resistant system, proving only a support for the floor slab.
Hence, their design is essentially based on gravity load combination cases. Assuming that the
design storey masses include also an allowance for seismic live-load (as mentioned in design
data section 1.2) and that they are uniform distributed in the storey area, the reference static
scheme can be represent as a simply supported beam with a triangular load acting.
45
Chapter 3. Design of Prototype Structure
Considering the maxima values present in the respectively diagrams, the flexural and shear
design foresee the actions listed in
Table 3.8
accurately specified both for storey and roof beam.
Table 3.8 Gravity beam design action
Beam Type
Action Type
Beam type
Symbol
Values
Unit
Storey beam
q storey
35.8
[ KN/m ]
Roof Beam
q roof
25.5
[ KN/m ]
107.3
[ KN ]
Load for unit length
Storey beam
V
0
B,max
V
0
B,max
Shear
GRAVITY BEAM
76.6
[ KN ]
Storey beam
Mbeam,1
286.0
[ KN m ]
Roof Beam
Mbeam,1
204.3
[ KN m ]
Roof Beam
Moment
In this case both the selected shape sections correspond to standardized steel profile: IPE 330
for the storey gravity beams (UNI 5398, EU 19) and IPE-A 300 for the roof gravity beams.
As in the previous cases, the actual design and the estimation of the percentage difference
between the required strength and the strength capacity are summarized in the following
tables.
Table 3.9 Selected Shape profile for gravity beam sections
Beam group
h
bf
tw
tf
r
A
IX
2
4
Wel,x
Wpl,x
3
3
IY
Wel,y
4
3
Wpl,y
3
[-]
[ mm ]
[ mm ]
[ mm ]
[ mm ]
[ mm ]
[ cm ]
[ cm ]
[ cm ]
[ cm ]
[ cm ]
[ cm ]
[ cm ]
Storey Gravity Beam
330.0
160.0
7.5
11.5
18.0
62.6
11766.9
804.3
713.1
788.1
98.5
153.8
Roof Gravity Beam
297.0
150.0
6.1
9.2
15.0
46.5
7173.5
483.1
541.8
519.0
107.4
69.2
Table 3.10 Percentage difference between flexural strength demand and flexural strength capacity
Demand
Capacity
Percentage difference
Beam group
MCD,X
MCD,X
MCD,X
[-]
[ KN m ]
[ KN m ]
[%]
Storey Gravity Beam
286.0
309.7
8.26
Roof Gravity Beam
204.3
208.6
2.10
3.3 Design Considerations
Recalling the research’s aim to verify the consistency of direct displacement-based design
methodology, the design is successful in so far as the effective strength capacity of each
structural element has been closed to the design indications. Therefore, the proposed project is
46
Chapter 3. Design of Prototype Structure
essentially based on flexural design. For the same reasons, only some preliminary
verifications are performed in order to validate the designed structure. If any negative
response was found in the verification checks, this does not invalidate the entire design, but,
just offers some hints for more general considerations.
The first preliminary verification regards the necessary columns’ flexural strength to contrast
seismic biaxial attack. In fact, there will be equal probability that the maximum seismic input
will occur in any orientation with respect to the principal axes. This means that the
development of plastic hinges mechanism is expected also simultaneously in both the
principal directions. Consequently, since the columns should remain essentially elastic after
the development of beam plastic-hinge mechanisms, they have been provided sufficient
diagonal strength capacity. This requirement appears more urgent if arranged in the specific
structural layout adopted, where the same column is usually part of two-way seismic frames.
Referring to the general case illustrate in Figure 3.6 , the required sum of column diagonal
moment capacities measured at the joint centroid has to respect the following inequality
[Priestley et al, 2007]:
∑M
CD
≥
(M B1P + M B1N )2 + (M B 2 P + M B 2 N )2
(3.2)
where
MCD
MBiP and MBiN
is the column diagonal moment capacity;
indicate the positive and negative beam moments;
Figure 3.6 Plan view of moment input for biaxial attack to two-way frame interior column
It is clear that for a I-shape steel column, where marked differences distinguish the capacity
strength owned in one direction with respect to the other (strong vs weak axis), this
prescription is extremely restrictive and difficult to satisfy in a reasonable design context.
Therefore, avoiding inefficacious complexities, in the design of prototype structure the biaxial
attack was taking into consideration amplifying of about 40% the modulus of seismic actions
in the different direction. Precisely, recalling the simplification valid for symmetrically
47
Chapter 3. Design of Prototype Structure
reinforced square columns in a two-way frames [Priestley et al,2007], a factor equal to 2 is
introduce in the equation (2.34).
In the reliable design of a two-way steel frame structure, steel sections characterized by equal
strength capacity in both the principal direction have to be preferred. This is the case of box or
hallowed structural sections but also of “austrian” cross-shape sections, as suggested by
Mazzolani [Mazzolani et al., 2006]. The austian cross-shape is obtained coupling two I-shape
profile trough industrial welding, as shown in Figure 3.7.
Figure 3.7 Austrian cross-shape section
The fidelity to the original structural layout imposed however the maintenance of I-shape
sections for steel columns, even if this lead to the selection of very thick element similar to
those indicated as HD European steel class.
3.4 Closing remarks regarding the design of prototype structure
An opportune combination of the orthogonal seismic actions guarantees a detail design for all
the structural elements: walls, columns and beams. The design procedure will be essentially
based on the satisfaction of flexural strength requirements.
In particular, the wall design is governed by the design maximum curvature strongly
influencing both the amount and spacing of transverse and longitudinal reinforcement. The
frame design is, instead, governed by the satisfaction of flexural strength simultaneously in
both the principal directions. This lead to a little adjustment in the original structural layout:
the internal columns’ axes have been subjected to a rotation of ninety degrees, therefore their
local section strong axis will now lies on global transverse direction (Figure 3.5).
48
Chapter 4. Verification of Numerical Structural Model
4 VERIFICATION OF NUMERICAL STRUCTURAL MODEL
Once completely determined the design of the prototype structure, an accurate definition and
verification of numerical structural models represents the next essential stage for the
development of reliable static and dynamic non linear analysis. Therefore, finite element
models of the prototype structure are realized using software programs such as SeismoStruct
[v. 4.0.9 built 992] and SAP2000 [v.10.0.1 advantage].
In particular, based on SAP2000 results, a sensitive analysis is carried out in order to calibrate
and validate two different SeismoStruct models. In fact, even if referred to the same case
study structure, these two models differ essentially for the strategy adopted to introduce the
pinned connections, the beam-end restrained that strongly characterized the original layout of
the structure.
A briefly introduction to Seismostruct and SAP2000 [v.10.0.1 advantage] software features,
will help to highlight and discern their most important peculiarities and differences.
4.1 SAP and SeismoStruct
SeismoStruct [v. 4.0.9 built 992] is a finite elements package capable of predicting the large
displacement behaviour of space frames under static or dynamic loading, taking into account
both geometric nonlinearities and material inelasticity.
Geometric nonlinearities play a fundamental role in the global response of the structure when
the occurrences of large deformation in the structural elements induce displacements not more
proportional to the loads effectively applied. Involving both local and global aspects, three are
the most important sources of geometric nonlinearities: the beam-column effects, the large
displacement/rotation effects and the P-delta effects.
Figure 4.1 Local chord system [SeismoStruct, 2007]
With the employment of a co-rotational formulation for the large displacement/rotation and a
cubic formulation for the beam-column effects, the secondary order effects are automatically
consider in the SeismoStruct program. With regard to the large displacement/rotation, a local
49
Chapter 4. Verification of Numerical Structural Model
chord system is attached to each finite element. Firmly following the element movements
(translation and rotation), this local reference system is able to described the current unknown
deformation and tension state of each individual element. (see Figure 4.1). The final
transformation of element’s internal forces and stiffness matrix obtained in the local chord
system, into the global coordinates system allows then the large displacements/rotations to be
accounted in the global response of the structure [Oran, 1973; Izzuddin, 1991].
In the second case, the beam-column effects, a cubic formulation by Izzudin [1991]
completely described the phenomenon, evaluating the transverse displacement as function of
the end-rotations of the element.
Crucial aspect for the correct definition of the system’s non linear response, the material
inelasticity is modelled extending the inelastic behaviour to the whole element trough the
fibre element methodology. This particular approach foresees the subdivision of each element
into a fixed number of elementary segments with the border sections following the NavierBernoulli approximation (plan sections remain plane). The element response is then evaluated
by numerical integration of nonlinear uniaxial stress-strain response of each individual fibres
in which the section has been subdivided.
Figure 4.2 Fibre element model[SeismoStruct, 2007]
On the other site, in SAP2000 [v.10.0.1 advantage] computer code the material inelasticity is
introduced by the user’s definition of high-plasticity zones, usually known as plastic hinges
zones. According to this approach, each element is essentially characterized by an elastic
behaviour with exception for these particular zones where all the deformations are considered
to be concentrated.
The differences between SeismoStruct and SAP2000 [v.10.0.1 advantage] regard not only the
material inelasticity but also the geometric nonlinearities such as P-delta and large
displacement effects. These tools, for example, are not default standard settings but are
available only for some specific analysis such as non linear direct-integration time-history
analysis and only if specifically required by the user. Moreover, the material nonlinearity is
not considered at all in the code and a little library is offered to define the different material
types. In fact in SAP2000 [v.10.0.1 advantage] can be employed only elastic materials
characterized by isotropic, orthotropic or uniaxial behaviour; while, on the contrary,
50
Chapter 4. Verification of Numerical Structural Model
SeismoStruct code [v. 4.0.9 built 992] disposes of a vast gallery counting eleven material
types (elastic, linear, bilinear, nonlinear, .etc). For these reasons, the SAP is used only in the
first phase of the numerical study, where typically elastic analyses (eigenvalue analyses) are
foreseen. The SAP2000 [v.10.0.1 advantage] results are, then, used to calibrate and validate
the SeismoStruct models, destined to performed nonlinear analysis such as static pushover
and dynamic time- history analysis.
In the following paragraphs will be mentioned the most peculiar assumptions adopted in both
the computer programs, SeismoStruct [v. 4.0.9 built 992] and SAP2000 [v.10.0.1 advantage],
for the modelling of the 3D case study structure.
4.2 SeismoStruct models
Exploiting the tools available in SeismoStruct [v. 4.0.9 built 992], two different models are
built in order to differently simulate the presence of pinned connections. As mentioned in
chapter 1 and chapter 2, in fact all the beams connected to the wall system are characterized
by pin-ends. In particular, the beams laying in transverse direction can be considered as
simply supported at both ends and subjected only to gravity load, inducing not any seismic
actions either in the reinforced concrete walls or in the steel columns. On the contrary, in the
longitudinal direction, even if no moment will be transmitted to the flanges wall by the pinned
beam ends, the seismic shear in the beam will induce moments at the centre line of the walls
reducing the base moment demand in the channel weak-axis direction. Therefore deeply
influencing the seismic interaction between frame and wall systems, the correct modelling of
pinned connections assumes a rule of primary importance. For this reason, two different
modelling tools are exploited and compared: the link element tools and the nodal constraint
tools.
4.2.1 Modelling consideration
Starting from the common aspects, a complete description of the three-dimensional structural
models is next proposed. The main features presented in detailed include the material
descriptions, the 3D layout scheme, the floor modelling, the mass discretization, the global
mass activated direction and the simulation of pinned connections presence.
(a)
Material
All the elements are defined as 3D inelastic beam-column element, capable of capturing
geometric and material nonlinearities considering 200 section fibres each. The material
properties used in the model are:
(a) Non linear constant confinement concrete model
(b) Menegotto-pinto steel model
Figure 4.3 Stress-Strain model for the structural materials adopted in SeismoStruct
51
Chapter 4. Verification of Numerical Structural Model
(1)
Nonlinear constant confinement concrete model (con_cc): The confined and
unconfined concrete is modelled using a unified stress-strain model based on the
formulation initially proposed by Mander [Mander et al,1988] for a concrete subjected
to uni-axial compressive loading and confined by transverse reinforcement (see Figure
4.3, a). The following mechanical properties are defined: compressive strength
fc=39000 kPa; tensile strength ft=3000 kPa; the strain at peak stress εc=0.002 mm/mm;
the confinement factor is assumed as 1.2 for confined concrete and as 1 for the
unconfined one; the specific weight is set equal to 0 kN/m3 since the masses are
manually assigned.
(2)
Menegotto-Pinto Steel model (stl_mp): The uni-axial steel model based on the stressstrain relationship proposed by Menegotto and Pinto [1973] is select to model both the
structural and the reinforcing steel (Figure 4.3, b). Except for the different yield
strength (assumed as fy=385 Mpa for structural steel and equal to fy=440 Mpa for
reinforcing steel), the indication of all the other parameters results common for both:
modulus of elasticity Es=200 GPa, strain hardening parameter μ=0.005 and, as in the
previous case, specific weight equal to 0 kN/m3.
The element’s formulation
The formulation of the element determines whether the element are based on displacement
shape functions (stiffness- or displacement based- element) or interpolation function for
forces (flexibility- or force-based element). The consideration of the element type is important
since it controls the distribution of the inelastic strains. Therefore the outcome of the analysis
will strongly depend on the chosen element formulation, the number and position of
integration points along the element length.
In SeismoStruct [v. 4.0.9 built 992], the distributed inelasticity frame elements are
implemented with the displacement-based (DB) finite elements formulations. In this case,
cubic Hermitian polynimials are used as displacement shape functions, corresponding for
instance to a linear variation of curvature along the entire element’s length. Since the
curvature field can be highly nonlinear during inelastic analysis such as push-over or inelastic
dynamic time history, a refined discretization (meshing) of the structural element (typically 45 elements per structural member) is required with a DB formulation. Adopting this
shrewdness, in fact, the assumption of a linear curvature field inside each of the sub-domains
does not prevent to capture the real deformed shape of the structure since the curvature is not
continuous across nodes.
(b)
(c)
3D layout scheme
Characterized by the section’s properties defined in chapter 3, three dimensional
displacement-based finite elements define completely the building structural skeleton,
modelling the walls, columns and beams actually present in the prototype structure.
The beams and the columns are modelled as steel elements with I-shape profile. All the frame
elements are modelled from centreline node to centreline node, without the use of any specific
elements to represent beam-column joint. This modelling approach, besides fully matches the
assumption made during the design phase, represents also a common practice in numerical
tests. In fact also in advantage structural analyses, is quite usual to neglect the beam-column
joints interaction given the uncertainty associated with the appropriate stiffness of beamcolumn joints and the minor effect that their inclusion have on the overall building response.
Considering the existence of stiff wall elements which act in parallel to the frames, this
52
Chapter 4. Verification of Numerical Structural Model
omission appears also more negligible in the prototype structure’s numerical models
[Sullivan, 2007].
Figure 4.4 Numerical model’ structural layout .
Concerning the RC core structure, the “wide-column analogy” (known also as the “equivalent
frame method”) has been adopted to model the complex force distribution between the
different components of non-planar walls (webs and flanges). In WCMs of non-planar walls
the web and the flange sections are represented by vertical column elements located at the
centroid of the web and flange sections. These vertical elements are then connected by
horizontal links running along the weak axis of the sections having common nodes at the
corners (see Figure 4.5).
The WCMs analogy requires the subdivision of U-shaped sections into three rectangular
planar wall sections, i.e. the web and the flanges. The corner areas were half attributed to the
web section and half to the flange sections, while the definition of reinforcing bars within the
rectangular concrete sections follows as established in Chapter 3.
Horizontal rigid
links
Vertical elements
representing web
and flanges
Figure 4.5 Model scheme used to represent U-shape wall system [Beyer et al., 2008]
53
Chapter 4. Verification of Numerical Structural Model
Inserted to represent the planar wall sections through the connection with all the vertical
elements, the horizontal links are modelled as rigid assigning infinite flexure and shear
stiffnesses as suggested by Reynouard and Fardis [2001].
As underlined by many scientists, the vertical spacing of the horizontal links influences the
behaviour of the WCM in two major aspects. First of all, allowing an effective
compatibility with respect to the axial elongations and rotations sustained by the flanges
and the web. In fact, establishing a direct connection between two elements, this coherence
in the undergone displacement field is enforced at the effective link’s locations. Secondly,
the spacing of the horizontal link influences also the magnitude of the parasitic bending
moment which occur as a consequence of the transmission of shear forces from the links to
the wall elements (Beyer [2008]). The larger the spacing of the links the larger the parasitic
bending moments introduced into the wall elements. Therefore, in the SeismoStruct
numerical models the link spacing is set equal to the storey height as a common and
consolidated practice suggests (e.g. Xenidis and Avramidis [1999]).
The WCMs has been adopted following the aim of reproducing the three-dimensional
configurations of the U-shape walls present in the structural layout. However the use of the
data provided in chapter 3 for the RC elements, may provide a level of flexural strength
higher than how intended. In fact should be recall that, in order to promote a rapid design
process, the core structure has been ideally split into three distinct elements: the central
web and the two lateral flanges. Each of ones has been design as a cantilever wall perfectly
able to resist alone to the design solicitations acting on its plan. Therefore using these
design data, the moulding of core structures could create a structural system globally
characterized by a higher magnitude of lateral flexural resistance, due to the collaboration
established between the three singular components.
Figure 4.6 General 3D view of SeismoStruc model
The development of a simple, efficient and computational inexpensive analysis models is
of primary importance in this project where inelastic analysis characterized by complex
displacement or acceleration field are applied to a three-dimensional model. For this reason
the number of element between two consecutive nodes is reduced only to one unit, even if
for displacement-based elements this is not appropriate. As mentioned in section (b), in
54
Chapter 4. Verification of Numerical Structural Model
fact, the curvature is linear along the entire length of DB elements and, in the case of
strongly nonlinearly curvature variation, this assumption may lead to not very accurate
solutions. However, even if this drawback could affect the analysis outcomes, the
numerical models are perfectly able to predict the global seismic behaviour of the entire
prototype structure. Therefore, the refinement in the number of element between two
consecutive nodes appears as not crucial problem for the research’s objective, destined to
be solved in a second phase when the most important verifications will have been already
performing.
In the Figure 4.6 is depicted the final configuration assumed by SeismoStruc model where
the beam, column and wall assembly perfectly matches the original structural
configuration.
(d)
Floor modelling
Stated as initial hypothesis, the rigid floor condition is realized imposing rigid diaphragm
constraints at each level of the structure. All the joints lying in the same floor level are
linked each other by special connections working as rigid links in the story plane but
allowing out-of-plane deformation (z-direction). Then, the joint relative displacements in
x-y parallel plane are not allowed, but remain fully guaranteed the out of plane flexibility
of the floor as theoretically established the rigid diaphragm behaviour.
Figure 4.7 Rigid diaphragm constraints
In SeismoStruct the rigid diaphragm tools requires the selection of a master node to define
the constraint net in the slab area. All the joints will be directly connected to it which
becomes the floor reference point for the software elaborations. Sensitivity analysis, based
on the possible master node locations, show as a central position has to be preferred.
Consequently, the geometrical barycentre of each floor is selected as rigid diaphragm
master node (see Figure 4.7). At each level, this point actually coincides also with the
centre of mass and stiffness of the storey due to the symmetry characterizing the entire
structure. Physically it can be identify as the mid-span point of the central beam in the
second transverse steel frame.
55
Chapter 4. Verification of Numerical Structural Model
Table 4.1 Beam and wall tributary masses
1
3
5
6
7
8
9
10
11
12
Mass
Beam
Type I:
Beam
Type II:
Beam
Type III:
Beam
Type IV:
Beam
Type V:
Beam
Type VI:
4 m Wall
8 m Wall
mi
m/l 1
m/l 2
m/l 3
m/l 4
m/l 5
m/l 6
m4m_WALL
m8m_WALL
[ - ]
[ t ]
[ t/m ]
[ t/m ]
[ t/m ]
[ t/m ]
[ t/m ]
[ t/m ]
[t]
[t]
12
11
10
9
8
7
6
5
4
3
2
1
0
500
700
700
700
700
700
700
700
700
700
700
770
0
2.60
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
4.01
0.00
1.30
1.82
1.82
1.82
1.82
1.82
1.82
1.82
1.82
1.82
1.82
2.01
0.00
1.95
2.73
2.73
2.73
2.73
2.73
2.73
2.73
2.73
2.73
2.73
3.01
0.00
0.65
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
1.00
0.00
2.28
3.19
3.19
3.19
3.19
3.19
3.19
3.19
3.19
3.19
3.19
3.51
0.00
0.65
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
1.00
0.00
10.0
14.0
14.0
14.0
14.0
14.0
14.0
14.0
14.0
14.0
14.0
15.4
0.00
39.9
55.9
55.9
55.9
55.9
55.9
55.9
55.9
55.9
55.9
55.9
61.5
0.00
Level
(e)
Mass discretization
Generally speaking, an excessive refinement in mass distribution assumption should be
avoided to not income in unfruitful computational efforts. However since a 2D or 3D
seismic input is considered, the correct representation of the mass torsional inertia
assumes a primary importance [Priestley et al, 2007]. Following these considerations, the
final models present the wall tributary mass concentrated at the each storey node of the
wall section and the beam tributary mass uniformly distributed along the entire element
length.
Referring to a uniform distribution of the storey mass on the floor surface and to the twoway slab structure, six beam type groups are defined considering the tributary area and
length associated to each element: the core beams (Type I), the lateral beams (Type II),
the corner beams (Type III), the inner beams (Type IV), the beams in front of 8 m walls
(Type V) and, finally, the gravity beam (Type VI). On the other side, the evaluation of the
tributary mass assigned to web and flanges walls is quite immediate: once determined the
wall system tributary area, the wall tributary mass is calculate in proportion to the element
section’s length (see fig in Appendix B).
In
Table 4.1 are summarized the storey tributary mass for unit length associated to each
beam group and the storey masses assigned to the 8 m walls and 4 m walls.
Avoiding time consuming model configurations, the wall masses are introduced in
SeismoStruct [v.4.0.9 built 992] as lumped mass element (displayed as green cube) while
the beam masses are expressed as additional mass/length feature in the beam’s section
properties tools.
(f)
Global mass direction
Focusing the attention on the dynamic seismic response of the structure, the wall lumped
masses are inserted in both principal direction x and y, while, the beam masses are
automatically inserted by the program since the additional mass/length feature is adopted.
It would be noticed that in SeismoStruct, there is the possibility of constraining the
dynamic degrees of freedom to only a few directions of interested. Exploiting this
opportunity, appropriate combinations of the global mass direction are activated in this
56
Chapter 4. Verification of Numerical Structural Model
study, depending on the peculiarity of the analysis performed (eigenvalue analysis,
pushover analysis or dynamic time history analysis). Without anticipate at this early stage
each combination adopted, an opportune recall will declare and justify the particular mass
directions selected in each analysis.
(g)
Simulation of pinned connection constraint
As mentioned section 4.2, in order to simulate the pinned connection presence, two
different modelling tools are exploited and compared: the link element tools and the nodal
constraint tools.
Figure 4.8 Link element independent degree of freedom
1) Pinned connection modelled as link element: In this case, 3D link elements with
uncoupled axial, shear and moment actions are introduce in the model. These link
elements connect two initially coincident structural nodes, and require the definition of an
independent force-displacement (or moment-rotation) response curve for each of its local
six degrees-of-freedom (F1, F2, F3, M1, M2, M3), as indicated in Figure 4.8.
Consequently, in order to model pinned joint conditions, linear response curves are
defined for all the six degrees-of-freedom. But, while very large stiffness values are
adopted for those degrees-of-freedom for which identical response of the two nodes is
expected, zero stiffness is associated to the uncoupled degrees-of-freedom. In that way no
relative displacement between the two extremities is allowed except for the uncoupled
degree-of-freedom. Moreover, it should be noticed that in order to avoid difficulties to
obtain the analysis’ convergence, instead of a strictly theoretical K=0, for the linear
response curve of the uncoupled degree-of-freedom a very small value of the stiffness is
preferred (i.e. 0.001).
2) Pinned connection modelled as nodal constaint type: In this case a connection between
the degrees-of-freedom of the two nodes converging at the same joint is introduced by
equal DOF constraint type. Belonging at the same tools, as in the case of the rigid
diaphragm constraint, the definition of a master node and its relative slave node is
compulsory. Once the correlation between the degrees-of-freedom of the slave node and
those of the master node is established, an appropriate setting of the displacement and
rotation restraint guarantees the possibility to simulate all kind of joint. In this case, all the
displacement and rotation restraints will be selected, excepted for the rotations that remain
released (around x-axis or y-axis) depending on the case considered.
In the Figure 4.9 the two SeismoStruct model are depicted. The presence of black cubes at the
pin- end location makes recognizable the model characterized by link element (part a) from
that characterized by nodal constraint type(part b). Moreover the green cubes at each floor
represent the wall lumped masses, the blue horizontal line the rigid diaphragm and finally the
grey cubes at the base nodes the fixed restraints.
57
Chapter 4. Verification of Numerical Structural Model
(a) Link element SeismoStruct model
(b) EDOF constraint SeismoStruct model
Figure 4.9 SeismoStruct models
4.3 SAP model
For consistency sake, the SAP model is built following the same modelling criteria adopted
for SeismoStruct models. In this way, is allowed not only a direct comparison between the
different results but also an opportune calibration of some peculiar parameters present in
SeismoStruct [v. 4.0.9 built 992] and usually set by the user (i.e. the penalty function
exponent).
Some aspects will however distinguish SAP2000 and SeismoStruct models such as the
definition of material properties, the modelling of pinned connection, the rigid diaphragms
and finally the rigid link element used to connect the beams to the wall element. The more
relevant aspects are therefore summarized in the following sections.
(a)
Materials:
Two different elastic materials are defined:
(1)
Uni-axial concrete model: for the walls, with a compressive strength fc=39000 kPa; a modulus
of elasticity EC=2.57x107 kPa and a specific weight γ =0 kN/m3. The analysis are carried out
both with and without the presence of reinforcement bars in the transverse sections, which is
assigned a yield strength equal to fy=440 Mpa.
(2)
Uni-axial steel model: for the structural steel element characterized by a yield strength
equal to fy=385 Mpa; a modulus of elasticity ES=200 GPa and a specific weight γ =0
kN/m3.
As can be noticed, no elastic section properties were assigned to the elements in SAP2000
model to properly compare the analyses’results with SeismoStruct models. In fact, the
calibration will take place on the basis of eigenvalue analyses, commonly influenced only by
the elastic section’s properties. Otherwise in normal circumstances, it has been emphasised
that the use of cracked section properties should be preferred in order to obtain reliable
results.
58
Chapter 4. Verification of Numerical Structural Model
(b)
Pinned connection and rigid diaphragm:
The pinned connection and the rigid diaphragm are modelled using the relative specific tools
available in SAP2000 [v.10.0.1 advantage]. In the first case, the pinned connections are
obtained by selecting the opportune boxes in the Frame Realise menu toolbars. On the other
side, the rigid diaphragms are defined by the use of Joint Constraints toolbars, collecting all
joints that lie in the same floor plan under the relative diaphragm constraint type. By default,
the diaphragm constraint causes all of its constrained joints to move together as a planar
diaphragm that is rigid against membrane (in-plane) deformation. No other parameters are
asked to be defined by the user.
(c)
Rigid link:
As in the previous case beams, columns and walls are modelled as finite elements defined
from node to node without any characterization of beam-column joint. The section properties
following the prescriptions indicated in chapter 3 and no particularity has to be mentioned.
Instead, the rigid link elements used to connect the beams to the wall element are modelled as
rigid beam with end offset properties activated along the entire length of the element. That
implies so high values for the section stiffness that the beam can be assumed as fully rigid.
The Figure 4.10 depicts the three dimensional SAP2000 model utilized for the analysis. In the
prospective view the orthogonal vector at the base of each floor wall sections represent the
wall storey masses, also in this case defined only in the x-y plane. Unfortunately, in SAP
program there is no the possibility to select or activate specific combination of global mass
directions: all the directions are automatically considered in the analyses.
Figure 4.10 SAP model
59
Chapter 4. Verification of Numerical Structural Model
4.4 Closing remarks regarding prototype structure’s numerical models
Two different models are built in SeismoStruct [v. 4.0.9 built 992] exploiting all peculiar
tools available in the software code. In fact, even if referred to the same case study structure,
these two models differ essentially for the strategy adopted to introduce the pinned
connections, the beam-end restrained that strongly characterized the original layout of the
structure. Both the SeismoStruct models will be then tested, calibrated and validated through
a direct comparison with a third model realized with the aid of SAP2000 [v.10.0.1 advantage]
computer code. For this reason, all the main modelling aspects have been presented and
briefly discussed. Particular attention has been given therefore to the material’s mechanical
properties, to the three dimensional layout scheme, to the floor modelling, to the mass
discretization and finally to the different methods to simulate the pinned connection
constraint.
4.5 Eigenvalue analysis
Assuming the more appropriate model configurations, three distinct models have been
defined: two in SeismoStruc [v. 4.0.9 built 992] and one in SAP2000 program [v.10.0.1
advantage]. Exploiting both the programs, the eigenvalue analysis is now carried out in order
to verify the efficiency of modelling choices.
Thirty vibration modes are then evaluated by each program to fully describe the dynamic
behaviour of the whole system. The results obtained give exhaustive information about the
seismic response of the entire structure, individualizing not only the natural periods but also
the mode shapes and modal participating masses.
As mentioned, this procedure allows not only a direct comparison between the three different
models but also an opportune calibration of those parameters present in SeismoStruct [v. 4.0.9
built 992] and usually set by the user (i.e. the penalty function exponent).
Due to the particular structural configuration that foresees the contemporary presence of RC
U-shape walls and steel frames, should be noticed how the entire nominal section stiffness is
adopted in the analyses not taking into account the possible effect of section cracking such as
the reduction of the second moment of area J or of the elastic modulus E. Moreover, no
damping is considered in the analyses, avoiding any problems in the comparison between the
different damping approaches adopted by software codes.
4.5.1 Eigenvalue analysis in SAP
The eigenvector modal analysis type is adopted in SAP2000 [v.10.0.1 advantage], considering
the model defined in section 4.3. It is observed that the presence or not of reinforcing bars in
RC wall section does not sensitively affect the results. For this reason in the following table
are shown only the results obtained considering the presence of reinforcing bars.
As a comment it is possible to notice that the usually code threshold of 85% for the
participating mass is reached just at the third mode for the x-direction and only at the 21th for
the y-direction. But comparing the effective mass percentages in the table list, it can be
observed as from 4th to 20th the vibration modes result or spurious or only z-direction
concerning. Therefore, the code threshold will be reasonably reached within the firsts four
60
Chapter 4. Verification of Numerical Structural Model
modes in both the principal directions, if, in some way, there was the possibility to neglect
these meaningless modes in SAP2000 code.
Table 4.2 Eigenvalue results for SAP model
SAP RESULTS
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1.468
1.070
0.289
0.218
0.205
0.182
0.179
0.174
0.174
0.171
0.167
0.165
0.163
0.156
0.156
0.144
0.144
0.143
0.143
0.140
0.128
0.112
0.078
0.078
0.077
0.067
0.067
0.067
0.066
0.060
67.33%
0.00%
17.89%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
6.91%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
65.31%
0.00%
0.00%
0.00%
0.00%
0.00%
0.08%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
20.11%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
36.34%
0.00%
0.00%
32.60%
0.00%
0.00%
10.31%
0.00%
0.00%
0.00%
0.74%
0.00%
0.00%
0.00%
1.79%
0.00%
0.00%
0.00%
0.00%
0.00%
2.27%
0.00%
0.00%
2.97%
0.00%
0.00%
2.83%
67.33%
67.33%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
85.22%
92.14%
92.14%
92.14%
92.14%
92.14%
92.14%
92.14%
92.14%
92.14%
0.00%
65.32%
65.32%
65.32%
65.32%
65.32%
65.32%
65.40%
65.40%
65.40%
65.40%
65.40%
65.40%
65.40%
65.40%
65.40%
65.40%
65.40%
65.41%
65.41%
85.52%
85.52%
85.52%
85.52%
85.52%
85.52%
85.52%
85.52%
85.52%
85.52%
0.00%
0.00%
0.00%
36.34%
36.34%
36.34%
68.95%
68.95%
68.95%
79.26%
79.26%
79.26%
79.26%
79.99%
79.99%
79.99%
79.99%
81.78%
81.78%
81.78%
81.78%
81.78%
81.78%
84.05%
84.05%
84.05%
87.02%
87.02%
87.02%
89.85%
4.5.2 Eigenvalue analysis in SeismoStruct
Defined as a purely elastic structural analysis, the eigenvalue analysis is carry on with elastic
material properties taken constant throughout the entire computation procedure. Even if
inelastic material types are defined in SeismoStruc models, the section’s elastic properties are
computed directly by the program depending on material type. For example in the case of
concrete material type, the modulus of elasticity equal to
EC = 4700 × f c0.5
(4.1)
is associated to an linear characterization of the material and the presence of longitudinal
reinforcement bars is taken into account. In the case of the prototype structure, the effective
computation of equation 4.1 gives as result:
EC = 2.57 × 10 7 kPa
61
Chapter 4. Verification of Numerical Structural Model
exactly the same value set in SAP2000 model for the concrete elastic modulus (paragraph
4.3), as consistency requirements ask.
As stated in section 4.2.1, the rigid diaphragms and the nodal constraints are introduced in
SeismoStruct program as penalty function algorithms characterized by penalty function
exponent. These exponents allow a directly calibration of the rigid link stiffness with respect
to that assumed by the whole complex of the structural element analysed. Their values are,
then, usually set by the user according to the structural behaviour to be match. In this case,
comparing the numerical results obtain with SAP2000 and SeismoStruct models, a sensitivity
analysis has been performed in order to scientifically define these coefficients’ modulus. In
particular, two are the valid combinations highlighted.
Figure 4.11 SeismoStruct Eigenvalue analysis scheme
The first combination foresees as possible modulus for the penalty function exponents values
equal to:
-
Rigid link weights: 107;
Rigid diaphragm weights: 1014;
providing equal numerical results as the use of Lagrange multiplier, the second constraint
algorithm available in SeismoStruct code. The second combination, instead; guarantees a very
close matched with SAP2000 [v.10.0.1 advantage] numerical results in the characterization of
the second vibration mode. In this case, the penalty function exponents assume values equal
to:
62
Chapter 4. Verification of Numerical Structural Model
Rigid link weights:108;
Rigid diaphragm weights:1017;
-
For clearness sake in the following paragraphs, combination (1) and combination (2) will
address respectively the first set and second set of penalty function exponents. Therefore,
considering both the SeismoStruct models (with link elements and equal DOF) and both the
combinations for the penalty function coefficients, four are the eigenvalue analysis carried on
using SeismoStruct code [v.4.0.9 built 992], as clarifies the scheme depicted in Figure 4.11.
In the following tables are listed the SeismoStruct’s outputs referred to the eigenvalue
analyses carried on with penalty function constraint algorithm. For completeness sake, the
results obtained with Lagrange multiplier constraint algorithm are however submitted in
APPENDIX B.
(A)
Link Element SeismoStruct Model:
Case A1:
Penalty function exponents set equal to:
Rigid link weights:107;
Rigid diaphragm weights:1014;
Table 4.3 Eigenvalue results for Link Element SeismoStruct model
SEISMOSTRUCT
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1.345
1.118
0.883
0.282
0.211
0.163
0.107
0.078
0.060
0.056
0.040
0.031
0.034
0.025
0.023
0.019
0.017
0.017
0.013
0.013
0.012
0.010
0.009
0.009
0.008
0.007
0.007
0.007
0.007
0.006
67.13%
0.00%
0.00%
17.72%
0.00%
0.00%
6.66%
0.00%
0.00%
3.46%
0.00%
0.00%
2.05%
0.00%
1.27%
0.00%
0.79%
0.00%
0.00%
0.47%
0.00%
0.26%
0.00%
0.00%
0.13%
0.00%
0.00%
0.05%
0.01%
0.00%
0.00%
65.98%
0.00%
0.00%
18.72%
0.00%
0.00%
6.77%
0.00%
0.00%
3.49%
0.00%
0.00%
2.06%
0.00%
0.00%
0.00%
1.28%
0.00%
0.00%
0.79%
0.00%
0.00%
0.47%
0.00%
0.00%
0.26%
0.00%
0.00%
0.13%
-
67.13%
67.13%
67.13%
84.85%
84.85%
84.85%
91.51%
91.51%
91.51%
94.97%
94.97%
94.97%
97.02%
97.02%
98.29%
98.29%
99.08%
99.08%
99.08%
99.54%
99.54%
99.80%
99.80%
99.80%
99.93%
99.93%
99.93%
99.98%
100.00%
100.00%
0.00%
65.98%
65.98%
65.98%
84.70%
84.70%
84.70%
91.47%
91.47%
91.47%
94.95%
94.95%
94.95%
97.01%
97.01%
97.01%
97.01%
98.29%
98.29%
98.29%
99.07%
99.07%
99.07%
99.54%
99.54%
99.54%
99.80%
99.80%
99.80%
99.92%
-
63
Chapter 4. Verification of Numerical Structural Model
Case A2:
-
Penalty function exponents set equal to:
Rigid link weights:108;
Rigid diaphragm weights:1017;
Table 4.4 Eigenvalue results for Link Element SeismoStruct model
SEISMOSTRUCT
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1.246
1.054
0.861
0.281
0.211
0.163
0.107
0.078
0.060
0.056
0.040
0.031
0.034
0.025
0.023
0.019
0.017
0.017
0.013
0.013
0.012
0.010
0.009
0.004
0.004
0.006
0.007
0.009
0.008
0.005
67.16%
0.00%
0.00%
17.69%
0.00%
0.00%
6.66%
0.00%
0.00%
3.46%
0.00%
0.00%
2.05%
0.00%
1.27%
0.00%
0.79%
0.00%
0.00%
0.47%
0.00%
0.26%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.13%
0.00%
0.00%
65.93%
0.06%
0.00%
18.70%
0.00%
0.00%
6.77%
0.00%
0.00%
3.49%
0.00%
0.00%
2.06%
0.00%
0.00%
0.00%
1.28%
0.00%
0.00%
0.79%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.47%
0.00%
0.00%
-
67.16%
67.16%
67.16%
84.86%
84.86%
84.86%
91.51%
91.51%
91.51%
94.97%
94.97%
94.97%
97.02%
97.02%
98.29%
98.29%
99.08%
99.08%
99.08%
99.54%
99.54%
99.80%
99.80%
99.80%
99.80%
99.80%
99.80%
99.80%
99.93%
99.93%
0.00%
65.93%
65.99%
65.99%
65.99%
84.69%
84.69%
91.47%
91.47%
91.47%
94.95%
94.95%
94.95%
97.01%
97.01%
97.01%
97.01%
98.28%
98.28%
98.28%
99.07%
99.07%
99.07%
99.07%
99.07%
99.07%
99.07%
99.54%
99.54%
99.54%
-
Differently from SAP2000 [v.10.0.1 advantage], in SeismoStruct [v. 4.0.9 built 992] is
realized the possibility to exclude vibration modes only z-direction interesting constraining
the global mass only to few directions of interest (namely X,Y and RZ).This leads to a drastic
reduction of spurious modes that have no structural meaning or interest, as testify Table 4.5
and Table 4.6.
Limited are the differences that distinguish the two output table: negligible or absent in the
estimation of participating masses and percentage difference lower than 10% in the evaluation
of modal period for the principal vibration modes. In both the cases, the 90% of the total
modal participating mass is reached considering just the firsts three modes for each direction.
(B)
Equal DOF SeismoStruct Model:
Penalty function exponents set equal to:
Case B1:
Rigid link weights:107;
Rigid diaphragm weights:1014;
64
Chapter 4. Verification of Numerical Structural Model
Table 4.5 Eigenvalue results for Equal DOF SeismoStruct model
SEISMOSTRUCT
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1.570
1.123
0.893
0.369
0.212
0.165
0.147
0.061
0.078
0.078
0.048
0.040
0.031
0.032
0.025
0.023
0.019
0.018
0.017
0.014
0.013
0.012
0.012
0.010
0.009
0.009
0.009
0.007
0.007
0.006
68.91%
0.00%
0.00%
16.23%
0.00%
0.00%
6.46%
0.00%
0.00%
3.40%
2.03%
0.00%
0.00%
1.26%
0.00%
0.78%
0.00%
0.46%
0.00%
0.26%
0.00%
0.00%
0.13%
0.05%
0.00%
0.00%
0.01%
0.00%
0.00%
0.00%
0.00%
65.95%
0.00%
0.00%
18.74%
0.00%
0.00%
0.00%
6.77%
0.00%
0.00%
3.49%
0.00%
0.00%
2.06%
0.00%
0.00%
0.00%
1.28%
0.00%
0.00%
0.79%
0.00%
0.00%
0.00%
0.47%
0.00%
0.00%
0.26%
0.12%
-
68.91%
68.91%
68.91%
85.15%
85.15%
85.15%
91.61%
91.61%
91.61%
95.01%
97.04%
97.04%
97.04%
98.30%
98.30%
99.08%
99.08%
99.55%
99.55%
99.80%
99.80%
99.80%
99.93%
99.99%
99.99%
99.99%
100.00%
100.00%
100.00%
100.00%
0.00%
65.95%
65.95%
65.95%
84.69%
84.69%
84.69%
84.69%
91.46%
91.46%
91.46%
94.95%
94.95%
94.95%
97.01%
97.01%
97.01%
97.01%
98.29%
98.29%
98.29%
99.07%
99.07%
99.07%
99.07%
99.54%
99.54%
99.54%
99.80%
99.92%
-
As in the previous case, the same observation can be noticed: negligible or absent differences
in the estimation of participating masses; a percentage difference lower than 10% in the
evaluation of modal period for the principal vibration modes. The 90% of the total
participating mass is reached considering just the firsts three modes in each direction.
Penalty function exponents set equal to:
Rigid link weights:108;
Rigid diaphragm weights:1017;
Case B2
-
Table 4.6 Eigenvalue results for Equal DOF SeismoStruct model
SEISMOSTRUCT
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
1
2
3
4
5
6
7
1.433
1.068
0.833
0.367
0.211
0.164
0.147
68.65%
0.00%
0.03%
16.47%
0.00%
0.00%
6.46%
0.00%
65.95%
0.00%
0.00%
18.73%
0.00%
0.00%
-
68.65%
68.65%
68.68%
85.15%
85.15%
85.15%
91.61%
0.00%
65.95%
65.96%
65.96%
84.68%
84.68%
84.68%
-
65
Chapter 4. Verification of Numerical Structural Model
SEISMOSTRUCT
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0.061
0.078
0.078
0.048
0.040
0.031
0.032
0.025
0.023
0.019
0.018
0.017
0.013
0.014
0.012
0.012
0.009
0.004
0.004
0.006
0.010
0.009
0.009
0.00%
0.00%
3.40%
2.03%
0.00%
0.00%
1.26%
0.00%
0.78%
0.00%
0.46%
0.00%
0.00%
0.26%
0.00%
0.13%
0.00%
0.00%
0.00%
0.00%
0.05%
0.01%
0.00%
0.00%
6.78%
0.00%
0.00%
3.49%
0.00%
0.00%
2.06%
0.00%
0.00%
0.00%
1.28%
0.00%
0.00%
0.79%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.47%
-
91.61%
91.61%
95.01%
97.04%
97.04%
97.04%
98.30%
98.30%
99.08%
99.08%
99.55%
99.55%
99.55%
99.80%
99.80%
99.93%
99.93%
99.93%
99.93%
99.93%
99.99%
100.00%
100.00%
84.68%
91.46%
91.46%
91.46%
94.95%
94.95%
94.95%
97.01%
97.01%
97.01%
97.01%
98.28%
98.28%
98.28%
99.07%
99.07%
99.07%
99.07%
99.07%
99.07%
99.07%
99.07%
99.54%
-
4.5.3 Comparison between SeismoStruct and SAP
In order to facilitate a direct comparison between SeismoStruct and SAP2000 [v.10.0.1
advantage] results, the summarizing Table 4.7 has been compiled, considering only the firsts
vibration modes in the two principal direction, the longitudinal or x- direction and the
transverse or y-direction.
Table 4.7 Eigenvalues comparison between SAP and SeismoStruct
1
2
Mode
SAP2000
3
4
SEISMOSTRUCT
LINK ELEMENT
5
6
SEISMOSTRUCT
Equal DOF
7
8
9
10
Modal Period Differences between
SAP2000 and SeismoStruct model
case A1
case A2
case B1
case B2
case A1
case A2
case B1
case B2
[ sec ]
[ sec ]
[ sec ]
[ sec ]
[ sec ]
[%]
[%]
[%]
[%]
1
ST
X-DIR
1.468
1.345
1.246
1.570
1.433
-8.38
-15.12
+6.95
-2.38
1
ST
Y-DIR
1.070
1.118
1.054
1.123
1.068
+4.49
-1.49
+4.95
0.19
2
ND
X-DIR
0.289
0.282
0.281
0.369
0.367
-2.42
-2.77
+27.68
+26.99
2
ND
Y-DIR
0.128
0.211
0.163
0.212
0.211
+64.84
+27.34
+65.63
+64.84
3
RD
X-DIR
0.112
0.107
0.107
0.147
0.147
-3.90
-3.90
+27.34
+27.34
Neglecting the infinitesimal discrepancies for the evaluation of participating masses, the
attention is then focused on the estimation of modal periods. Distinguished code by code and
case by case, the principal modal periods are listed from Col. 2 to Col. 6, while the respective
66
Chapter 4. Verification of Numerical Structural Model
percentage differences between SAP2000 and SeismoStruct code are listed from Col. 7 to
Col. 10.Some conclusions can be point out:
-
For each SeismoStruct models, can be observed a clear decrease in the estimation of
the vibration periods, passing from case 1 to case 2. Therefore for the same structure,
an effective increment of the overall lateral stiffness can be obtained simply
amplifying the magnitude of penalty function exponents. However, the differences
remain circumscribed to the firsts modes in both the principal direction, while seem to
not affect the higher modes.
-
Despite the different tools exploited, is ascertained a clear consistency between the
link element and the equal DOF SeismoStruct models, how testifies the close match
between the different sets of eigenvalue outputs.
-
There is also a clear consistency between SAP2000 and SeismoStruct models. In
particular with respect to the SAP2000 solutions, the values indicated by SeismoStruct
link element model generally represent a lower band limits while that proposed by
equal DOF constitute an upper band limits. The differences remain however restricted
not exceeding 0.22 sec. A comparison between the mode shapes further ratifies the
consistency obtained.
-
A more accurate outputs’ analysis shows how the SeismoStruct model B2 (equal
DOF) best matches the peculiar features indicated by SAP2000 model for the
principal vibration modes in both x and y direction. The higher modes are, instead,
better capture by both A1 and A2 (link element) models.
-
Carrying on the sensitive analysis, is noticed that the link element SeismoStruct
models results more stable than the equal DOF model with respect to any modelling
choices (i.e., the activation of global mass direction and the variation of penalty
function exponent magnitude). In fact, sometimes computational problems incurred in
the equal DOF model’s analyses, making difficult and expensively time-consuming
the results achievement.
-
In order to obtain results consistent with SAP2000 outputs, the global mass directions
that can be activated in SeismoStruct are X, Y, Z and RZ, excluding, therefore, the
rotational contribution in RY and RX directions. Moreover, if the vertical direction Z
remains unselected, the possibility to reduce spurious vibration modes is realized
excluding that modes only z-direction interesting.
-
The penalty function exponents are the modelling coefficients that more influence the
dynamic response of the overall structure. In fact, considering different combination of
those parameters, not always the SAP2000 and SeismoStruct outputs result
comparable. Therefore, the future analysis will be carried out setting up only the two
combinations previously mentioned.
4.6 Closing remarks regarding the verification of numerical structural models
The two different SeismoStruct models (equal DOF and link element model) have been
successfully calibrated on the bases of SAP2000 model results.
67
Chapter 4. Verification of Numerical Structural Model
During the sensitivity analysis performed in SeismoStruct, clearly emerges the importance of
a correct configuration for the penalty function exponents, parametric coefficients referred to
the modelling constrain condition present in the numerical model such as rigid diaphragm,
link element, etc. Named case 1 and case 2, two are the possible combinations addressed for
these coefficients that more influence the global response of the entire structure.
4.7 Modal deformed shapes
1st Mode in Longitudinal Direction
1st Mode in Transverse Direction
2nd Mode in Longitudinal Direction
2nd Mode in Transverse Direction
3rd Mode in Longitudinal Direction
3rd Mode in Transverse Direction
Figure 4.12 Modal deformed shape: pure translational modes
68
Chapter 4. Verification of Numerical Structural Model
1st Torsional Mode
2nd Torsional Mode
3rd Torsional Mode
4th Torsional Mode
Figure 4.13 Modal deformed shapes: torsional modes
Despite the use of different numerical models, the vibration modes are presented always with
the same succession order in all the cases analysed. Observing, in fact, the eigenvalues’
results proposed from Table 4.3 to Table 4.6, the pure vibration modes in longitudinal
direction, occupy in the output list always the first, the fourth and the seventh positions while
the second and the fifth are designated to pure vibration modes in transverse direction and the
third, sixth and the eighth to the torsional ones. The respect of this sequence in each
eigenvalues analysis highlights the stability and the reliability of each numerical model
adopted. Moreover, is also underlined the extremely efficiency of SeismoStruct computer
code [v. 4.0.9 built 992] able to capture and numerically translate each different modelling
features adopted, strongly maintaining constant and stable solutions.
In Figure 4.12 and Figure 4.13 Errore. L'origine riferimento non è stata trovata.are,
therefore, presented the firsts modal deformed shapes assumed with respect to pure
translational and roto-translational vibration modes.
69
Chapter 5. Design Verification through Pushover and Nonlinear Time History
5 DESIGN VERIFICATION THROUGH PUSHOVER AND
NONLINEAR TIME HISTORY ANALYSIS.
Performing several studies on the prototype structure numerical models, a detailed
characterization of the global seismic response is offered. For this purpose, two different types
of analysis are exploited: inelastic pushover analysis and inelastic dynamic time history
analysis (IDTHA). The pushover analyses are performed in order to define the effective
overstrength factor characterizing the dynamic response of the structure. The dynamic time
history analyses are, instead, adopted to verify the actual response of the prototype structure
under seismic load.
5.1 Pushover analysis
In non linear static analysis, different pattern of horizontal loads are applied to structural
models in order to simulate the actual distribution of inertial forces during the earthquake
motion. The task of these horizontal forces is to “push“ the structure into the inelastic
behaviour till reached the collapse condition. For this reason, the non linear static analyses are
also famous with the name of “pushover analysis”.
Once defined the load pattern and maintaining unvaried the relative proportion between them,
the horizontal loads are progressively amplified in order to monotonically increase the
horizontal displacement of a selected control point, usually localized at the building top level.
Step by step, a “capacity curve“ can be trace plotting the progressive displacement of the
control point ΔD as function of the base shear Vbase experienced by the system. Finally, to
allow the passage from the real MDOF system to the equivalent SDOF system, the capacity
curve obtained is then properly scaled adopting a participating factor Γ. Directly related with
the first mode vibration shapes, the participating factor Γ is calculated as:
Γ=
∑m φ
∑m φ
i i
2
i i
(5.1)
The chart represented in Figure 5.1 clearly illustrates the procedure previously described. In
this way, the base shear seismic demand can be directly evaluate in accordance to the
maximum displacement expected for the equivalent SDOF system in the specific limit state
considered.
70
Chapter 5. Design Verification through Pushover and Nonlinear Time History
3.6E+04
MDOF
System
Base Shear V base [ KN ]
3.2E+04
2.8E+04
Γ
2.4E+04
2.0E+04
Equivalent
SDOF
Vbase
1.6E+04
1.2E+04
8.0E+03
4.0E+03
0.0E+00
0
0.5
Δ Dc
1
1.5
2
2.5
3
Displacement Δ c [ m ]
Figure 5.1 Capacity curve example
5.1.1 Horizontal lateral load pattern
The Italian code OPCM 3431 suggests the adoption of two different horizontal distributions
to perform non linear static analyses: the “uniform” pattern and the “modal” pattern. The first
is related with the hypothesis of inertial forces proportional to the mass distribution, while the
second foresees horizontal seismic loads as proportional to the lateral displacement obtained
in the first mode of multimodal (elastic) analysis. Coherently with the design assumption
adopted in this research, the “modal” pattern has been selected to carry out the pushover
analysis in both the principal directions.
In order to completely define the modal load patterns, different spreadsheets have been built
in relation to the particular direction analysed. In Table 5.1, for instance, is presented a
spreadsheet sample obtained considering the longitudinal direction (x-direction). Three are the
principal steps followed for the compilation of this table. First of all, the nodal displacements
characterizing the principal vibration mode are selected from the eigenvalue analysis (Col.4).
Then, the modal displacements are multiplied for the relative nodal mass and each product is
normalized respect to the entire sum of all the products (Col.6 and Col.7). Finally, the initial
set of lateral forces is obtained simply multiplying the results listed in Col.7 for an initial trial
value of lateral force, assumed arbitrarily equal to 10 kN.
Since each peculiar eigenvalue response has to be taking into account, the same procedure is
repeated for all the SeismoStruct numerical models analysed (link element and equal DOF )
and for all the penalty function exponent combinations defined (case 1 and case 2),
considering finally both the principal directions.
71
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Table 5.1 Spreadsheet sample of modal pattern distribution (longitudinal direction)
1
2
3
4
Storey
Node
Name
Mass
m
Displacement
ΔX
[-]
[-]
[ ton ]
[m]
13 Storey
n1113
n2113
n3113
n4113
n5113
n2213
n3213
n4213
n2313
n3313
n4313
n1413
n2413
n3413
n4413
n5413
nxxx1213
nx1213
nxx1313
nxx4313
nx5213
nxx4213
10.42
20.83
20.83
20.83
10.42
32.55
41.67
32.55
32.55
41.67
32.55
10.42
20.83
20.83
20.83
10.42
9.98
39.93
9.98
9.98
39.93
9.98
……
……
1 Storey
n112
n212
n312
n412
n512
n222
n322
n422
n232
n332
n432
n142
n242
n342
n442
n542
nxxx122
nx122
nxx132
nxx432
nx522
nxx422
th
st
5
ΔX
Normalized
φx
6
7
8
m*φx
m*φx
Normalized
Modal Force
Distribution
[m]
[ ton ]
[ - ]
[ KN ]
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
2.01E-05
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.04E+01
2.08E+01
2.08E+01
2.08E+01
1.04E+01
3.26E+01
4.17E+01
3.26E+01
3.26E+01
4.17E+01
3.26E+01
1.04E+01
2.08E+01
2.08E+01
2.08E+01
1.04E+01
9.98E+00
3.99E+01
9.98E+00
9.98E+00
3.99E+01
9.98E+00
2.81E-03
5.62E-03
5.62E-03
5.62E-03
2.81E-03
8.78E-03
1.12E-02
8.78E-03
8.78E-03
1.12E-02
8.78E-03
2.81E-03
5.62E-03
5.62E-03
5.62E-03
2.81E-03
2.69E-03
1.08E-02
2.69E-03
2.69E-03
1.08E-02
2.69E-03
2.81E-02
5.62E-02
5.62E-02
5.62E-02
2.81E-02
8.78E-02
1.12E-01
8.78E-02
8.78E-02
1.12E-01
8.78E-02
2.81E-02
5.62E-02
5.62E-02
5.62E-02
2.81E-02
2.69E-02
1.08E-01
2.69E-02
2.69E-02
1.08E-01
2.69E-02
……
……
……
……
……
……
16.04
32.08
32.08
32.08
16.04
50.13
64.17
50.13
50.13
64.17
50.13
16.04
32.08
32.08
32.08
16.04
15.37
61.49
15.37
15.37
61.49
15.37
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
4.33E-07
Maximum
Value
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
2.16E-02
Maximum
Value
3.46E-01
6.92E-01
6.92E-01
6.92E-01
3.46E-01
1.08E+00
1.38E+00
1.08E+00
1.08E+00
1.38E+00
1.08E+00
3.46E-01
6.92E-01
6.92E-01
6.92E-01
3.46E-01
3.32E-01
1.33E+00
3.32E-01
3.32E-01
1.33E+00
3.32E-01
9.33E-05
1.87E-04
1.87E-04
1.87E-04
9.33E-05
2.92E-04
3.73E-04
2.92E-04
2.92E-04
3.73E-04
2.92E-04
9.33E-05
1.87E-04
1.87E-04
1.87E-04
9.33E-05
8.94E-05
3.58E-04
8.94E-05
8.94E-05
3.58E-04
8.94E-05
9.33E-04
1.87E-03
1.87E-03
1.87E-03
9.33E-04
2.92E-03
3.73E-03
2.92E-03
2.92E-03
3.73E-03
2.92E-03
9.33E-04
1.87E-03
1.87E-03
1.87E-03
9.33E-04
8.94E-04
3.58E-03
8.94E-04
8.94E-04
3.58E-03
8.94E-04
Sum
Sum
Sum
2.01E-05
1.00
3.71E+03
1.00
10.00
72
Chapter 5. Design Verification through Pushover and Nonlinear Time History
5.1.2 Static Pushover analysis in SeismoStruct
Once defined the initial pattern, the magnitude of lateral force is progressively increased
maintaining unalterated the relative load’s proportions during the entire development of
pushover analysis.
In SeismoStruct this amplification can be obtained, for example, through the adoption of the
Response control strategy. Guarantying the respect of convergence criteria, a direct increment
in the displacement of the control node is imposed and the correspondent load factor is
numerically evaluated at each step. This load factor represents, in fact, the numerical
coefficient necessary to apply to the load pattern in order to obtain the imposed displacement.
The procedure is automatically iterated until a determinate limit, structural thresholds or
numerical failures are reached.
Within the context of performance-based design, a primary importance is assumed by the
identification of the exact instance in which the different performance limit states are reached.
For this reason in the SeismoStruct code, some performance criteria are introduced in order to
evaluate the rise and the development of non-structural damages, structural damages and
collapse states occurred. In particular, two are the principle collapse mechanism take into
consideration by the code:
-
the crush of core concrete material: that can occur in compression states when the
material strains result larger than the ultimate crushing strain threshold assumed equal
to -0.005;
-
the fracture of steel reinforcement bar: that can occur in tensile states when the steel
strains result larger than the fracture strain threshold assumed equal to +0.060;
When one of the previous performance criteria is exceeded during the analyses, a warning line
instantaneously informs the user declaring both the collapse mechanism verified and the value
reached by the incremental load factor. Is then possible to individuate the exact instant in
which each singular member failure occurred with respect to the system’s pushover curve. In
this way, the local element response is related to the global response of the entire structure
and possible global failure criteria can be argued.
5.2 Verification of the Displacement–Based Designed Structure through Pushover
Analisis
Besides offering important feedbacks on the efficiency of the design, the principal scope of
nonlinear static analysis is to estimate the overstrength presented by the structure. For this
reason in accordance to the design displacement ΔD, a direct comparison between the design
base shear and the actual base shear recorded in the nonlinear test is investigated.
Considering the various case study, sixteen pushover analyses are carried out in order to
evaluate the inelastic behaviour of the structure. Unfortunately, some of them incur in
unknown computational error making impossible the achievement of reliable results. The
problems principally affect the numerical model B2 characterized by the use of equal DOF
constraint. In these cases the nomenclature N.A. (not available) is introduced in the
summarizing tables next shown.
Due to the unknown computational errors affecting the equal DOF models (the B models), the
link element numerical models (the A models) will be considered in the dynamic time history
analysis, offering more probabilities of stable solutions.
73
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Table 5.2 SeismoStruct link element model: pushover analysis results.
Penalty Function
combination 1
Design Values
Penalty Function
combination 2
Design SDOF
Displacement
SDOF
effective
period
Effective
mass
Effective
Stiffness
Base
Shear
Base
Shear
X POS DIR
Base
Shear
X NEG DIR
Base
Shear
X POS DIR
Base
Shear
X NEG DIR
ΔD
Te
me
Ke
Vbase
Vbase
Vbase
Vbase
Vbase
[m]
[ sec ]
[ ton ]
[ MN/m ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
0.479
2.788
6371.9
32.36
15514.4
17004.2
17009.8
17019.8
17029.2
-
-
-
-
-
9.60%
9.64%
9.70%
9.76%
(a) Longitudinal direction (x-direction)
Penalty Function
combination 1
Design Values
Penalty Function
combination 2
Design SDOF
Displacement
SDOF
effective
period
Effective
mass
Effective
Stiffness
Base
Shear
Base
Shear
Y POS DIR
Base
Shear
Y NEG DIR
Base
Shear
Y POS DIR
Base
Shear
Y NEG DIR
ΔD
Te
me
Ke
Vbase
Vbase
Vbase
Vbase
Vbase
[m]
[ sec ]
[ ton ]
[ MN/m ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
0.476
2.996
6352.6
27.93
13300.4
16488
16488
15991
15989
-
-
-
-
-
23.97%
23.96%
20.23%
20.21%
(b) Transverse direction (y-direction)
Table 5.3 SeismoStruct equal DOF model: pushover analysis results.
Penalty Function
combination 1
Design Values
Penalty Function
combination 2
Design SDOF
Displacement
SDOF
effective
period
Effective
mass
Effective
Stiffness
Base
Shear
Base
Shear
X POS DIR
Base
Shear
X NEG DIR
Base
Shear
X POS DIR
Base
Shear
X NEG DIR
ΔD
Te
me
Ke
Vbase
Vbase
Vbase
Vbase
Vbase
[m]
[ sec ]
[ ton ]
[ MN/m ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
0.479
2.788
6371.89
32.36
15514
15663
N.A.
N.A.
N.A.
-
-
-
-
-
9.60%
-
-
-
(a) Longitudinal direction (x-direction)
Penalty Function
combination 1
Design Values
Penalty Function
combination 2
Design SDOF
Displacement
SDOF
effective
period
Effective
mass
Effective
Stiffness
Base
Shear
Base
Shear
Y POS DIR
Base
Shear
Y NEG DIR
Base
Shear
Y POS DIR
Base
Shear
Y NEG DIR
ΔD
Te
me
Ke
Vbase
Vbase
Vbase
Vbase
Vbase
[m]
[ sec ]
[ ton ]
[ MN/m ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
0.476
2.996311
6352.637
27.93
13300.4
15464
N.A.
N.A.
N.A.
-
-
-
-
-
13.99%
-
-
-
(b) Transverse direction (y-direction)
The results listed in the previous tables testify a very good agreement between the actual
overstrength capacity presented by the structure and the overstrength capacity evaluated at the
74
Chapter 5. Design Verification through Pushover and Nonlinear Time History
final phase of design procedure (Table 3.4). For the link element models, in fact, the gap
between the design base shear and the actual base shear for the design displacement ΔD not
exceeds 10% in the longitudinal direction and 24% in the transversal one. More limited
appear the difference recorded in the equal DOF models, but the not availability of all the
data makes meaningless any projection.
2.80E+04
Real Structure
MDOF
Base Shear V
base
[ KN ]
2.40E+04
2.00E+04
1.60E+04
Vbase
Equivalent
SDOF
1.20E+04
8.00E+03
4.00E+03
0.00E+00
0
0.2
0.4
ΔD
0.6
0.8
1
1.2
Displacem ent Δ c [ m ]
Figure 5.2 Capacity curves obtained performing pushover analysis
Relating the MDOF structure to the equivalent SDOF system, in Figure 5.2 are depicted the
capacity curves obtained from pushover analyses. In particular, this chart is referred to the
numerical model B1 considering the longitudinal direction.
5.2.1 Closing remarks on Push-over analyses outcomes
Performing non linear static analyses, the efficiency and coherence of prototype structure’s
design has been verified and proved with respect to the initial design hypotheses. In
particular, the results indicate a great match between the design base shear and the maximum
shear resistance at base level.
During the pushover analyses, the rise of some computational errors in the equal DOF models
imposes the adoption of link element models as unique numerical models able to guarantee
stable solutions for the following analyses, the dynamic time-history analysis.
5.3 Dynamic Time History Analysis
Even if the vertical distribution of lateral forces is calibrated on modal analysis’ results, the
pushover analysis remains however a static analysis unable to simulate higher mode effects on
the structural response. Therefore, the inelastic time history analysis represents the most
accurate method for verifying nonlinear inelastic response of a structure subjected to
earthquake loading: the inelastic deformations and rotation can be accurately evaluate and
investigated the influence of higher modes effects.
75
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Exploiting SeismoStruct tools, inelastic time history analyses are carried out considering the
link element models and both the combinations for penalty function exponents (case A1 and
case A2).
Applied at each base node, a ground acceleration time-history simulates the dynamic input
typical of seismic motion. Focusing the attention on the actual response of the prototype
structure, the dynamic soil-structure interaction has been neglected and only the horizontal
component of the ground motion has been considered. The IDTHA have been carried out
considering both the principal directions, the ground accelerations has been then applied
firstly considering the transverse direction alignment (y-direction) and secondly the
longitudinal one (x-direction). In that way, a direct comparison between the actual seismic
performance and the adopted design hypothesis can be accurately performed.
5.3.1 Dynamic input
As prescribed in Eurocode 8 (section 3.2.3.1), both artificial and real accelerograms recorded
can be used in dynamic time-history analysis to simulate seismic input.
1.10
Design Acceleration Response Spectrum
1.00
EQK_1_Acceleration Response Spectrum
Pseudo-Acceleration SA [ g ]
0.90
EQK_2_Acceleration Response Spectrum
EQK_3_Acceleration Response Spectrum
0.80
EQK_4_Acceleration Response Spectrum
0.70
EQK_5_Acceleration Response Spectrum
EQK_6_Acceleration Response Spectrum
0.60
EQK_7_Acceleration Response Spectrum
EQK_15_Acceleration Response Spectrum
0.50
0.40
0.30
0.20
0.10
0.00
0
1
2
3
4
5
6
7
Period T [ sec ]
Figure 5.3 Design and artificial earthquakes’ Acceleration Response Spectra
1.50
1.40
1.30
Displacement S D(T) [ m ]
1.20
1.10
1.00
0.90
0.80
0.70
Displacement Design Spectrum
0.60
EQK_2_Displacement Spectrum
EQK_1_Displacement Spectrum
0.50
EQK_3_Displacement Spectru,
0.40
EQK_4_Displacement Spectrum
EQK_5_Displacement Spectrum
0.30
EQK_6_Displacement Spectrum
0.20
EQK_7_Displacement Spectrum
EQK_15_Displacement Spectrum
0.10
0.00
0
1
2
3
4
5
6
7
Period T [ sec ]
Figure 5.4 Design and artificial earthquakes’ Displacement Response Spectra
76
Chapter 5. Design Verification through Pushover and Nonlinear Time History
In the first case, the artificial records can be directly obtained using special purpose programs
taking into account all the initial conditions, while, in the second case, the real records have
been properly adapted to best match the design spectrum over the full range of period or at
least in the range of interest [Priestley et al, 2007].
With the aid of SIMQKE program [Carr, 2001], two set of seven artificial dynamic inputs
have been generated guaranteeing a fully compatibility with the design spectrum as shown in
Figure 5.3 and Figure 5.4. The seismic motion is, then, represented as a time-history record
where the accelerations are expressed as function of the time (Figure 5.5). As commonly used,
the entire input length is set equal to 20 sec.
Artificial Earthquake Record
EQK 1
Artificial Earthquake Record
EQK2
0.4
0.3
0.3
0.2
Acceleration [ g ]
Acceleration [ g ]
0.4
0.1
0
-0.1
-0.2
-0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
12
14
16
18
-0.4
20
0
2
4
6
8
Time [ sec ]
0.4
0.3
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
14
16
18
20
14
16
18
20
14
16
18
20
14
16
18
20
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
Time [ sec ]
10
12
Time [ sec ]
Artificial Earthquake Record
EQK5
Artificial Earthquake Record
EQK6
0.4
0.4
0.3
0.3
Acceleration [ g ]
Acceleration [ g ]
12
Artificial Earthquake Record
EQK4
0.4
Acceleration [ g ]
Acceleration [ g ]
Artificial Earthquake Record
EQK3
0.2
0.1
0
-0.1
-0.2
-0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
Time [ sec ]
10
12
Time [ sec ]
Artificial Earthquake Record
EQK7
Artificial Earthquake Record
EQK15
0.4
0.4
0.3
0.3
Acceleration [ g ]
Acceleration [ g ]
10
Time [ sec ]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
Time [ sec ]
10
12
Time [ sec ]
Figure 5.5 Artificial record acceleration time-histories
77
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Dynamic considerations suggest that the characterization based only on acceleration timehistory and on acceleration or displacement response spectra, is not sufficient to capture the
peculiar features of each different dynamic input. Therefore, is considered necessary the
evaluation of the respective Fourier amplitude spectra. Through this study, should be possible
to predict which artificial earthquake will result most severe or which will activate in major
measure the higher mode effects on the structure. Beyond the scope of this research, a
theoretical and rigours treatment is avoided but simple considerations are proposed and some
conclusions point out.
In Figure 5.6, for example, three Fourier amplitude spectra are depicted in a semi-logarithmic
chart with the frequency’s range comprised between 0.1 Hz to 100Hz. Even if generated by
the same code and respecting the same compatibility conditions, the three artificial records
EQK1, EQK2 and EQK15 appear clearly separate and distinct in the range of interest (0.4Hz 2.0 Hz). A numerical study based on the Fourier amplitudes assumed with respect to the
different modal periods, can demonstrate how EQK1 and EQK2 records result most severe
than EQK15, strongly exciting both fundamental and higher mode components.
EQK 1
EQK 2
EQK 15
4.5
4.0
Fourier Amplitude
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.1
1
10
100
Frequency [ Hz ]
Figure 5.6 Fourier Amplitude Spectra
Knowing the huge variations that can characterize different input ground motions in the
frequency’s content, Eurocode 8 (section 4.3.3.4.3) suggests to consider a set of seven or at
least three earthquake records to perform IDTHA. In the first case, the average of the seven
different response quantities obtained will represent the actual design response quantity, while
only the most severe values will be considered in the second case.
Adopting the first approach, a sensitivity analysis has been carried on to investigate the
variation in the response values depending on the particular group of earthquakes selected.
Therefore, two different sets of artificial record are defined: the first group will collect the
ground motions from EQK1 to EQK7 while the second from EQK2 to EQK15.
78
Chapter 5. Design Verification through Pushover and Nonlinear Time History
5.3.2 Inelastic dynamic time history in SeismoStruct
In SeismoStruct, the dynamic time history analysis foresees the use of Hilbert-Hughes-Taylor
(HHT) integration scheme to provide a direct integration of motions’ equations. Its peculiar
parameter are respectively set equal to α=-0.1, β=0.3025 and γ=0.6. Moreover, the same timestep considered in the acceleration time-histories is adopted (dt=0.01 sec) for consistency
sake. Based on an iterative procedure, the analysis settings required the definition of some
others convergence criteria and iterative parameters. A displacement tolerance equal to 0.001
m and a rotation tolerance equal to 10-3 rad are, for example, adopted as convergence criteria
while the iterative strategy foresees:
-
a maximum number of iteration equal to 200;
a number of updates at each iteration for the tangent stiffness matrix equal to 150;
a maximum tolerance equal to 1e20;
a maximum step reduction coefficient equal to 0.001;
Even if a large number of iteration and stiffness matrix updates are allowed in the integration
scheme, the analyses performed show how the convergence is immediately reached within the
first two iterations. A clear sign of stability and convergence for the numerical models
adopted.
Finally, as mentioned in section 5.3, the dynamic seismic inputs are simulated by the use of
acceleration loading curves (accelerograms) applied at all the structural base nodes.
5.4 Verification of the Displacement–Based Designed Structure through DTHA
Since there is no possibility to predict the direction of an earthquake attack, the capacity
design adopted in the DDBD procedure is calibrated to ensure a controlled response whatever
direction the ground motion operates (Chapter 2). Therefore, exploiting orthogonal
alignments, dynamic time-history analyses are performed in both the principal direction,
allowing a direct comparison between the hypothetical and actual response of the prototype
structure.
Due to the vastness of results collected, in the following sections is retained opportune to
present in detail all the results obtained in the transverse direction and only the most
important in the longitudinal one. Others results can be also examined in APPENDIX C,
completing accurately the information field on DDBD verification.
The maximum displacements, drifts and forces recorded in frames and walls during the timehistory analyses are then considered and examined allowing important considerations on the
capacity method adopted.
5.4.1 Transverse direction: Displacement Profiles.
As mentioned in section 5.3, a sensitive analysis is performed considering two different
combinations of penalty function exponents and two different sets of dynamic input.
Compared with the target displacement profile (the red line with rhomb marker), the
maximum floor displacements recorded during time-history analyses are presented in Figure
5.7. Organized in rows and columns, the scheme summarizes the results obtained in all the
four cases analysed. In particular, the rows distinguish the first from the second set of
artificial earthquakes used and the columns differentiate the two penalty function’s
combinations considered.
79
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
Relative Height ( hi/H )
Relative Height ( hi/H )
Chapter 5. Design Verification through Pushover and Nonlinear Time History
0.6
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.2
0.4
0.6
Lateral Displacement [ m ]
0.8
0.0
1.0
st
(a) Dynamic input :1 set; Penalty combination: 1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.8
1.0
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.2
0.4
0.6
Lateral Displacement [ m ]
(b) Dynamic input : 1st set; Penalty combination: 2
Relative Height ( hi/H )
Relative Height ( hi/H )
0.6
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Lateral Displacement [ m ]
nd
(c) Dynamic input :2 set; Penalty combination:1
0.0
0.2
0.4
0.6
Lateral Displacement [ m ]
0.8
1.0
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.7 Displacement profile: comparison between DDBD hypothesis and DTHA maximum response
It should be noticed that the maximum displacements profiles are built selecting the
maximum displacement recorded at each floor of the prototype structure and considering the
whole time-history output record. Therefore, the displacement profiles shown are actually
never assumed by the structure but represent an upper bound limit able to include the effects
induced by higher modes [Sullivan, 2007].
In order to make easier the comparison between the DDBD displacement profile and the
design response one, the Figure 5.8 has been prepared following the same logic adopted for
the previous chart. In each graph, the DDBD target displacements profile (the red line with
square marker) is compared with the design response profile depicted as blue line with square
marker. Following the Eurocode indications, the design response profile is calculated as the
average of the maximum response quantities obtained in the seven time-history analysis
performed.
80
Chapter 5. Design Verification through Pushover and Nonlinear Time History
p
11
10
10
9
9
8
8
7
7
Level
12
11
Level
12
6
6
5
5
4
4
3
3
2
2
1
1
0
0.00
0.20
DDBD Displacement
0.40
0.60
0
0.00
0.80
DTHA Displacement Response
0.20
DDBD Displacement
Displacement [ m ]
11
11
10
10
9
9
8
8
7
7
Level
Level
12
6
6
5
5
4
4
3
3
2
2
1
1
0.20
0.40
0.60
0.80
(b) Dynamic input : 1st set; Penalty combination: 2
12
DDBD Displacement
0.60
DTHA Displacement Response
Displacement [ m ]
(a) Dynamic input :1st set; Penalty combination: 1
0
0.00
0.40
0.80
DTHA Displacement Response
Displacement [ m ]
(c) Dynamic input :2nd set; Penalty combination:1
0
0.00
0.20
DDBD Displacement
0.40
0.60
0.80
DTHA Displacement Response
Displacement [ m ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.8 Diplacement profile: comparison between DDBD hypothesis and DTHA avarege response
In all the four cases, can be easily recognized an excellent match between the DDBD
displacement profile and the design response displacement. Except for the first and second
floors, the maximum percentage difference between the target and the response profile does
not exceed a 9%, as shown in Table 5.4. Even if the results are extremely satisfactory for all
the cases analysed, some observations and distinctions should be notice:
-
With the maximum difference recorded for the first (35-38%) and second (9%-13%)
floors, the DDBD displacement profile tends to overestimate the displacement
demands in the lower storeys of the building. While a light underestimation affect the
higher zone of the structure above the 5th storey.
81
Chapter 5. Design Verification through Pushover and Nonlinear Time History
-
Comparing the charts column by column, can be noticed how an increment in the
penalty function exponents’ magnitude provides to reinforce the influence of walls
behaviour in the global response of the prototype structure. Consequently, the
displacement profile tends to linearize the shape assumed.
-
Comparing the charts row by row can be, instead, proved the influence of different
ground motions in the final response quantities. In this case, the selection of two or
just one severe input loads in the set of artificial motions (see section 5.3.1) does not
affect very much the average design response of time-histories analysis, that oscillates
in a very limited range. However some differences can be clearly noticed: comparing
with DDBD design displacement profile, the results obtained with the second set are
almost coincident with a maximum percentage difference equal to 4.5%, while the use
of first set implies higher discrepancies with a maximum percentage difference equal
to 8.5% (not considering, in both the cases, the first and second floors);
Table 5.4 Displacement profiles: comparison between DDBD hypothesis and DTHA avarege response
Level
DDBD
Design
Displ.
DTHA
DTHA
DTHA
DTHA
Percentige Percentige Percentige Percentige
Response Response Response Response
Difference Difference difference difference
Case a
Case b
Case c
Case d
ΔDi
ΔAVERAGE
ΔAVERAGE
ΔAVERAGE
ΔAVERAGE
Case a
Case b
Case c
Case d
[-]
[m]
[m]
[m]
[m]
[m]
[ - ]
[ - ]
[ - ]
[ - ]
12
0.705
0.715
0.735
0.693
0.713
-1.5%
-4.2%
1.6%
-1.1%
11
0.645
0.660
0.676
0.639
0.655
-2.4%
-4.9%
0.8%
-1.7%
10
0.584
0.605
0.619
0.585
0.599
-3.6%
-5.9%
-0.2%
-2.5%
9
0.524
0.549
0.560
0.530
0.541
-4.7%
-6.8%
-1.2%
-3.2%
8
0.464
0.490
0.499
0.473
0.482
-5.7%
-7.6%
-1.9%
-3.8%
7
0.404
0.429
0.437
0.413
0.421
-6.2%
-8.2%
-2.4%
-4.3%
6
0.343
0.365
0.373
0.351
0.359
-6.4%
-8.5%
-2.4%
-4.5%
5
0.283
0.300
0.306
0.288
0.295
-5.8%
-8.1%
-1.7%
-3.9%
4
0.225
0.234
0.239
0.224
0.230
-4.0%
-6.3%
0.1%
-2.2%
3
0.167
0.167
0.171
0.161
0.164
0.1%
-2.2%
4.1%
1.8%
2
0.112
0.102
0.104
0.098
0.100
9.5%
7.4%
13.1%
11.1%
1
0.060
0.039
0.040
0.038
0.038
35.0%
33.5%
37.7%
36.2%
0
0.000
0.000
0.000
0.000
0.000
0.0%
0.0%
0.0%
0.0%
5.4.2 Transverse direction: Maximum Storey Drift.
At each instant of the time history, the storey drift are evaluated following the equation 5.1:
δ i (t ) =
where:
δi
di
di-1
hi
d i − d i −1
hi
(5.2)
is the interstorey drift
is the displacement at the floor i ;
is the displacement at the floor i-1 ;
is the height of storey i;
Following the same criteria stated in section 5.4.1, during the entire time-history analysis the
maximum value recorded is consider as the maximum storey drifts for each level. Then, the
82
Chapter 5. Design Verification through Pushover and Nonlinear Time History
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
design response maximum storey drifts is calculated as the average of the seven maximum
drifts profiles so obtained.
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
DTHA Drift
3
DDBD Drift
4
0
Code Drift
1
DTHA Drift
2
3
4
DDBD drift
Code Drift
Interstorey Drift [ % ]
Interstorey Drift [ % ]
(b) Dynamic input : 1st set; Penalty combination: 2
st
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
(a) Dynamic input :1 set; Penalty combination: 1
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1
DTHA Drift
2
3
DDBD Driftt
4
Code Drift
Interstorey Drift [ % ]
(c) Dynamic input :2nd set; Penalty combination:1
0
1
DTHA Drift
2
3
DDBD Drift
4
Code Drift
Interstorey Drift [ % ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.9 Drift profiles: comparison between DDBD hypothesis and DTHA average response
In Figure 5.9 are compared the design drift profile (represented as a red solid line), the code
drift limit (depicted as a black dashed line) and the design response maximum storey drifts
83
Chapter 5. Design Verification through Pushover and Nonlinear Time History
profile (marked as a blue solid line). A qualitative analysis shows how the global trend
addressed by design drift profile is respected by the design response maximum storey drifts
profile, even if some discrepancies can be observed:
-
an overestimation of first storey drift characterises all the cases examined, with
differences that reach 0.57 percentage units (see Table 5.5). With a lighter disparity
(about 0.13 percentage units), an overestimation of storey drifts affects also the higher
building’s levels.
-
on the contrary, an underestimation of drift storey can be noticed in the central part of
design profile, with differences that decrease increasing the height of the levels
considered, and, however, not grater than 0.38 percentage units;
-
The design approach has not been completely successful in limiting the storey drifts
since in three cases ( case a, case b and case d) the values recorded exceed not only the
DDBD drift profile but also the Eurocode drift limit.
-
Amplifying the penalty function exponents, the relative increment in the global system
stiffness tends to increase the distance between the predicted and the recorded drifts’
profile extending also the underestimation zone in the upper stories of the prototype
structure.
-
Also in this case, the selection of two or just one severe input load in the set of
artificial motions does not affect very much the average design response of timehistories analysis.
Table 5.5 Drift profiles: comparison between DDBD hypotheses and DTHA response
DTHA
DTHA
DTHA
DTHA
Response Response Response Response Percentige Percentige Percentige Percentige
Drift
Drift
Drift
Drift
Difference Difference Difference Difference
Case a
Case b
Case c
Case d
Code
Drift
Limit
DDBD
Design
Drift
δlimit
δD
δaverage
δaverage
δaverage
δaverage
[-]
[%]
[%]
[%]
[%]
[%]
[%]
[ - ]
[ - ]
[ - ]
[ - ]
12
11
10
9
8
7
6
5
4
3
2
1
0
2
2
2
2
2
2
2
2
2
2
2
2
2
1.88
1.88
1.88
1.88
1.88
1.88
1.87
1.84
1.79
1.72
1.63
1.51
1.51
1.81
1.83
1.88
1.94
1.99
2.03
2.06
2.08
2.08
2.05
1.96
0.98
0.98
1.90
1.92
1.96
2.02
2.06
2.09
2.12
2.12
2.13
2.10
2.00
1.00
1.00
1.77
1.79
1.83
1.89
1.93
1.97
2.00
2.01
2.01
1.97
1.88
0.94
0.94
1.86
1.88
1.92
1.97
2.01
2.03
2.05
2.05
2.05
2.02
1.92
0.96
0.96
0.07%
0.05%
0.01%
-0.05%
-0.11%
-0.14%
-0.19%
-0.24%
-0.29%
-0.33%
-0.33%
0.53%
0.53%
-0.02%
-0.04%
-0.08%
-0.13%
-0.18%
-0.21%
-0.25%
-0.29%
-0.34%
-0.38%
-0.37%
0.50%
0.50%
0.12%
0.10%
0.05%
0.00%
-0.05%
-0.08%
-0.13%
-0.17%
-0.22%
-0.25%
-0.25%
0.57%
0.57%
0.03%
0.01%
-0.04%
-0.08%
-0.12%
-0.15%
-0.18%
-0.21%
-0.26%
-0.30%
-0.29%
0.55%
0.55%
Level
Case a
Case b
Case c
Case d
5.4.3 Transverse direction: Wall shear forces.
The maximum shear experienced by the wall system is now evaluated. In Figure 5.10 are
depicted, in particular, the maximum shear forces developed during the DTHAs along the
84
Chapter 5. Design Verification through Pushover and Nonlinear Time History
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
entire height of the 8 m walls. The red solid line in each chart recalls the shear capacity
envelope adopted in the design procedure and accurately defined in section 2.4.5.
The general trend assumed by time-history response curves matches very well the triangular
regular shape addressed by the shear capacity envelope, with a general reduction in strength
demand up to the height of the building. Moreover in each of the four cases, the scatter
observed is in general quite small for the different DTHA response curves analysed. This
large convergence can be ascribed to the particular sets of accelerograms employed: each
record has been generated in order to closely match the design spectrum adopted.
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
5000
10000
15000
0
20000
5000
12
12
11
11
10
10
9
9
8
8
7
7
6
15000
20000
(b) Dynamic input : 1st set; Penalty combination: 2
Level
Level
(a) Dynamic input :1st set; Penalty combination: 1
10000
Shear [ KN ]
Shear [ KN ]
6
5
5
4
4
3
3
2
2
1
1
0
0
0
5000
10000
15000
20000
Shear [ KN ]
(c) Dynamic input :2nd set; Penalty combination:1
0
5000
10000
15000
20000
Shear [ KN ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.10 Wall shear profile: comparison between capacity envelope and DTHA maximum response
85
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Even if the different earthquakes records does not significantly affect the global response of
shear profiles, some observations can be however noticed. The DTHAs performed with the
first set of dynamic input, for example, foresee an higher magnitude of the shear experienced
in the higher wall’s levels. While with the adoption of the second set of dynamic input, the
shear profile results fairly constant up to the 8th floor, as can be easily recognize in case c and
case d charts of Figure 5.11.
Design Shear
Design Shear
DTHA Shear Response
DTHA Shear Response
DDBD Shear
DDBD Shear
11
11
10
10
9
9
8
8
7
7
Level
12
Level
12
6
6
5
5
4
4
3
3
2
2
1
1
0
-5000
0
5000
10000
0
-5000
15000
0
st
(a) Dynamic input :1 set; Penalty combination: 1
Design Shear
Design Shear
DTHA Shear Response
DTHA Shear Response
12
11
11
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0.0
5000.0
10000.0
15000.0
Shear [ KN ]
nd
15000
DDBD Shear
12
Level
Level
10000
(b) Dynamic input : 1st set; Penalty combination: 2
DDBD Shear
0
-5000.0
5000
Shear [ KN ]
Shear [ KN ]
(c) Dynamic input :2 set; Penalty combination:1
0
-5000
0
5000
10000 15000 20000
Shear [ KN ]
nd
(d) Dynamic input :2 set; Penalty combination:2
Figure 5.11 Wall shear profiles: comparison between DDBD hypothesis and DTHA avarege response
86
Chapter 5. Design Verification through Pushover and Nonlinear Time History
In the previous schemes, in particular, the DTHAs average response shear profile and the
design shear capacity envelope are compared with the DDBD shear profile.
The great amplification induced on the shear profile by the capacity design adoptions appears
now completely justified and very well calibrated. In fact, not only the general trend is very
well addressed, but also the base shear is predicted with excellent precision. Is so verify the
importance and the efficacy of capacity design procedures to protect structural elements
against higher mode effects which induce marked increment on wall shear demand. For
further information, numerical results are also submitted in APPENDIX C.
Transverse direction: Wall moments.
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
5.4.4
6
5
5
4
4
3
3
2
2
1
1
0
-40000
10000
60000
110000
0
-40000
160000
Wall Moment [ KN m ]
(a) Dynamic input :1st set; Penalty combination: 1
12
12
11
11
10
10
9
9
8
8
7
7
6
5
4
4
3
3
2
2
1
1
10000
60000
110000
160000
Wall Moment [ KN m ]
(c) Dynamic input :2nd set; Penalty combination:1
60000
110000
160000
6
5
0
-40000
10000
Wall Moment [ KN m ]
(b) Dynamic input : 1st set; Penalty combination: 2
Level
Level
6
0
-40000
10000
60000
110000
160000
Wall Moment [ KN m ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.12 Wall moment profile: comparison between DDBD hypothesis and DTHA maximum response
87
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Considering the transverse structural wall, the maximum moments developed in DTHAs by
each earthquake record are represent in Figure 5.12, distinguishing, as usual, the four cases
analysed. In each graph is also depicted the moment profile obtained from DDBD procedure
as a green dashed line. Also in this case, a clean convergence in moment demand can be
observed considering the different maximum response curves. All the profiles tend, in fact, to
overlap the same trace with the only exception in case b and case d where the light blue
response curve is clearly detached from the others. This curve, in particular, corresponds to
the maximum moment profile obtaining with the dynamic input EQK5, and should be recall
the case b and case d are characterized by the use of the same combination for the penalty
function exponent (Combination 2).
Design Shear
DTHA Shear Response
DDBD Shear Profile
Design Strength
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
Design Shear
DTHA Shear Response
DDBD Shear Profile
Effective Design
6
6
5
5
4
4
3
3
2
2
1
1
0
-50000
0
50000
0
-50000
100000 150000
Wall Moment [ KN m]
0
(a) Dynamic input :1st set; Penalty combination: 1
150000
Design Shear
DTHA Shear Response
DDBD Shear Profile
Design Strength
12
12
11
11
10
10
9
9
8
8
7
Level
7
Level
100000
(b) Dynamic input : 1st set; Penalty combination: 2
Design Shear
DDBD Shear Profile
DTHA Shear Response
Effective Design
6
6
5
5
4
4
3
3
2
2
1
1
0
-40000
50000
Wall Moment [ KN m ]
10000
60000
110000
Wall Moment [ KN m ]
(c) Dynamic input :2nd set; Penalty combination:1
0
-50000
0
50000
100000
150000
Wall Moment [ KN m]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.13 Wall moment profile: comparison between DDBD and DTHA avarage response
88
Chapter 5. Design Verification through Pushover and Nonlinear Time History
As already noticed by Sullivan [Sullivan, 2007] selecting the absolute maximum moment
magnitude, a predominant shape can be recognized for wall bending moment curves:
characterized by a bulge in the upper two-thirds of the structure, the profile descends reducing
moment demand in lower storeys till the ground level, where a substantial increment can be
noticed again.
Besides, as already noticed in wall shear forces, appear evident how higher modes effects
strongly increase the moment demands at the ground storey and in the mid-high level of the
structure (6th -10th storey). In these two zone, neither the capacity envelope (red solid line)
seems to provide enough safety for design predictions, while results quite conservative in the
central zone (Figure 5.13). Even if the DTHA average response curve exceeds the capacity
envelope, the overcoming remain, however, quite contained as indicated in Table 5.6.
Table 5.6 Wall moment profile: comparison between DDBD hypotheses and DTHA response
Level
DDBD
Wall
Moment
Ms,i
DTHA Wall DTHA Wall DTHA Wall DTHA Wall
Percentige Percentige Percentige Percentige
Moment
Moment
Moment
Moment
Difference Difference difference difference
Case a
Case b
Case c
Case d
MAVERAGE
MAVERAGE
MAVERAGE
MAVERAGE
Case a
Case b
Case c
Case d
[-]
[ KN m ]
[ KN m]
[ KN m]
[ KN m]
[ KN m ]
[ - ]
[ - ]
[ - ]
[ - ]
12
10627
2268
2374
2253
2824
78.7%
77.7%
78.8%
73.4%
11
19128
15602
15645
15190
17040
18.4%
18.2%
20.6%
10.9%
10
27630
30133
30440
29186
33675
-9.1%
-10.2%
-5.6%
-21.9%
9
36131
38899
40604
37249
45252
-7.7%
-12.4%
-3.1%
-25.2%
8
44633
43579
45188
42191
50219
2.4%
-1.2%
5.5%
-12.5%
7
52799
42603
43744
41581
49623
19.3%
17.2%
21.2%
6.0%
6
58621
40699
41558
39879
49610
30.6%
29.1%
32.0%
15.4%
5
64442
39961
40891
39012
44560
38.0%
36.5%
39.5%
30.9%
4
70263
39309
40892
38656
49222
44.1%
41.8%
45.0%
29.9%
3
76085
42963
44791
41916
52380
43.5%
41.1%
44.9%
31.2%
2
81906
50835
52241
49886
55621
37.9%
36.2%
39.1%
32.1%
1
87727
58075
59087
57316
61277
33.8%
32.6%
34.7%
30.2%
0
87727
99622
101147
98093
107745
-13.6%
-15.3%
-11.8%
-22.8%
Even if the actions overcame the capacity design provisions, the simplified design adoption of
constant uniform longitudinal reinforcement along the entire height of the wall guarantee
plentifully protection for all the levels with the exception for the base one. This result can be
easily visualized comparing the constant strength capacity profile (the light blue dashed line)
with the moment capacity envelope (the red solid line) presented in each chart composing
Figure 5.13. The design hypothesis of equal moment strength has been, then, revealed as
extremely useful to prevent plastic deformation upon the base level in the numerical models,
even if can not certainly represent an actual design provision due to the excessively
conservative results and obviously economic disadvantages.
5.4.5 Transverse direction: Frame shear forces in outer columns.
The maximum shear force experienced by frame outer columns is now taken into
consideration. Figure 5.14 shows how the maximum shear recorded are fairly constant along
the entire height of the building, fully respecting the initial assumption made by DDBD
design procedure. Only two non-linearities can be noticed at the top and bottom of the
building. In both zones the shear profile is characterized by a reduction in shear demand, more
consistently at the ground floor while less pronounced at the top level. These two singularities
can be probably ascribed to the complex interaction between frame and wall system. In fact,
89
Chapter 5. Design Verification through Pushover and Nonlinear Time History
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
as mentioned in the previous section, a plastic hinge is developed at the at the wall base level,
attracting and concentrated a great amount of the global seismic actions. While in the upper
levels the wall remains essentially in the elastic range, allowing a major distribution of shear
forces. Consequently, a higher shear proportion is allocated to the frame columns.
6
5
5
4
4
3
3
2
2
1
1
0
0
0.0
200.0
400.0
600.0
800.0
0
1000.0
200
400
600
800
1000
Shear [ KN ]
Shear [ KN ]
(a) Dynamic input :1st set; Penalty combination: 1
(b) Dynamic input : 1st set; Penalty combination: 2
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
6
6
6
V
5
5
4
4
3
3
2
2
1
1
0
0
0
200
400
600
800
1000
Shear [ KN ]
(c) Dynamic input :2nd set; Penalty combination:1
0
200
400
600
800
1000
Shear [ KN ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.14 Frame shear profile: comparison between DDBD hypothesis and DTHA maximum response
90
Chapter 5. Design Verification through Pushover and Nonlinear Time History
In the following figure, the DTHA average shear response is represented as blue line and
compared with the DDBD shear profile (depict as dashed green curve) and the capacity shear
curves (the red solid line) evaluated considering the capacity design provisions stated in
section 2.4.5. Even if the design envelope is overcome, a very low difference generally
separates the two profiles. A percentage difference equal to 18% characterized, in fact, case
(a) and case (b), while just a 10% separate the DTHA response profile to design envelope in
case (c) and (d). The greater percentage difference in the first two cases indicates a major
sensibility of shear action to the particular dynamic input set selected.
y
Design Shear
DTHA Shear Response
DDBD Shear Profile
Design Shear
DTHA Shear Response
DDBD Shear Profile
12
12
11
11
10
10
9
9
8
8
7
Level
Level
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
200
400
600
800
1000
Shear [ KN ]
(a) Dynamic input :1st set; Penalty combination: 1
0
200
600
800
1000
Design Shear
DTHA Shear Response
DDBD Shear Profile
Design Shear
DTHA Shear Response
DDBD Shear Profile
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
400
Shear [ KN ]
(b) Dynamic input : 1st set; Penalty combination: 2
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
200
400
600
800
1000
Shear [ KN ]
nd
(c) Dynamic input :2 set; Penalty combination:1
0
200
400
600
800
1000
Shear [ KN ]
nd
(d) Dynamic input :2 set; Penalty combination:2
Figure 5.15 Frame shear profile: comparison between DDBD hypothesis and DTHA avarege response
91
Chapter 5. Design Verification through Pushover and Nonlinear Time History
5.4.6 Transverse direction: Frame moments in outer columns.
The Figure 5.16 illustrates the maximum moments recorded during time history analyses in
the outer columns of transverse frame. As in the case of shear forces, the maximum moment
profile is characterized by a fairly constant pattern interrupted only by two non-linearity at the
top and at the bottom level.
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
The following charts demonstrate how the global shape profile is quite insensitive to the
particular seismic input selected, while the magnitude of maximum moment recorded is
clearly influenced by the dynamic characteristics of input motion. Therefore a limited but not
negligible scatter between each maximum moment curves can be appreciated.
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
500
1000
0
1500
500
12
12
11
11
10
10
9
9
8
8
7
7
6
1500
(b) Dynamic input : 1st set; Penalty combination: 2
Level
Level
(a) Dynamic input :1st set; Penalty combination: 1
1000
Outer column Moment [ KN m ]
Outer columns Moment [ KN m ]
6
5
5
4
4
3
3
2
2
1
1
0
0
0
500
1000
1500
Outer column Moment [ KN m ]
(c) Dynamic input :2nd set; Penalty combination:1
0
500
1000
1500
Outer column Moment [ KN m ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.16 DTHA maximum moment curves for outer columns in transverse direction
92
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Design Moment Profile
Average Moment Profile
DDBD Moment Profile
Design Moment Profile
Average Moment Profile
DDBD Moment Profile
12
12
11
11
10
10
9
9
8
8
7
Level
Level
7
6
5
6
5
4
4
3
3
2
2
1
1
0
0
500
1000
0
1500
0
Outer columns Moment [ KN m ]
st
(a) Dynamic input :1 set; Penalty combination: 1
Design Moment Profile
500
Design Moment Profile
Average Moment Profile
DDBD Moment Profile
Average Moment Profile
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
1500
(b) Dynamic input : 1st set; Penalty combination: 2
DDBD Moment Profile
6
1000
Outer column moment [ KN ]
6
5
5
4
4
3
3
2
2
1
1
0
0
0
500
1000
1500
Outer column Moment [ KN m ]
nd
(c) Dynamic input :2 set; Penalty combination:1
0
500
1000
1500
Outer column Moment [KN m]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.17 Comparison between DDBD, capacity design and DTHA response moment profiles
5.4.7 Transverse direction: frame shear force and moment in inner column.
Summarizing the results obtained for the inner columns in transverse direction, a complete
characterization of the seismic actions experienced during DTHAS is given in Figure 5.18 and
Figure 5.19, considering both shear and moment demands. As usual in these graphs, the
93
Chapter 5. Design Verification through Pushover and Nonlinear Time History
DTHA average response profiles (in blue line) are directly compared with the relative DDBD
indication (dashed green line) and with the capacity design envelope (red line).
The DTHA average response shear and moment profiles result fairly constant along the entire
height of the structure with exception for the higher and lower zones where same anomalies
affect the uniform pattern. Neglecting the small reduction of strength demand at the top level,
the attention is immediately captured at the bottom zone where huge discrepancies arise.
In the case of shear actions, the DTHA base shear response curves tend to space out from the
shear capacity envelope to closely match the value addressed by DDBD procedure.
Completely opposite the situation at the second level, where a huge underestimation separate
the actual DTHA shear response from the capacity design predictions. In the upper storey the
DTHA shear profile tends to regularise the pattern, assuming a fairly constant trend very close
to that indicated by capacity envelope. For accurate quantifications, all the numerical values
are available in Table C.0.4 in APPENDIX C.
The high shear actions experienced by the columns in the first storey, became determinant for
the creation of elevated magnitudes in the DTHA average moment profile. At the first level,
in fact, the moment magnitude is so amply to exceed the capacity design envelope, traced as
red line in Figure 5.19. However, the actual strength capacity provided to the core column is
almost sufficient to satisfy the high moment demands, even if remain excessively
conservative for the upper storey, where the seismic actions induced minors solicitations. In
particular from the second to the top level, the DTHA moment demand remains within the
range marked by DDBD and capacity envelope profiles.
These non-linearity in the shear and moment profiles of inner columns can be interpreted
considering the global behaviour of the lateral resistant system subjected to seismic actions.
Summarizing the main stages, the evolution of structural interaction under earthquake loading
can be explained as follow.
As stated in the design procedure for transverse direction, a great proportion (equal to 60%) of
the entire base shear has been allocated to the wall systems. The capacity design foresees a
collapse mechanism characterized by the development of plastic hinges at the wall base,
where the shear actions result most severe. Considering the amplification both in shear and
moment demand due to higher mode effects, the length of plastic hinges presumably increases
respect to design previsions (LP =2.58 m, see section 2.4.2), extending its influence near to the
first storey level.
As known, beyond the yield, the behaviour of plastic hinge is characterized by a progressive
reduction in the effective stiffness depending on the ductility demand requested in plastic
phase. Therefore, when the yield conditions has been reached at the wall base, a modification
in the global stiffness layout takes place with a consequently redistribution of shear actions
among the lateral resisting elements.
94
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Design Shear
Design Shear
DTHA average response
DTHA average response
DDBD Shear Profile
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
DDBD Shear Profile
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
500
1000
1500
2000
0
500
1000
Shear [ KN ]
(a) Dynamic input :1st set; Penalty combination: 1
2000
(b) Dynamic input : 1st set; Penalty combination: 2
Design Shear
Design Shear
DTHA average response
Average Shear
DDBD Shear Profile
DDBD Shear Profile
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
1500
Shear [ KN ]
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
500
1000
1500
2000
Shear [ KN ]
(c) Dynamic input :2nd set; Penalty combination:1
0
500
1000
1500
2000
Shear [ KN ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.18 Frame shear profile: comparison between DDBD hypothesis and DTHA average response
95
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Design Moment Profile
DTHA average response
Design Moment Profile
DTHA average response
DDBD Moment Profile
Effective Design Strength
Effective Design Strength
DDBD Moment Profile
11
11
10
10
9
9
8
8
7
7
Level
12
Level
12
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1000
2000
3000
0
4000
Inner column Moments [ KN m ]
(a) Dynamic input :1st set; Penalty combination: 1
1000
2000
3000
4000
Inner column Moments [ KN m ]
(b) Dynamic input : 1st set; Penalty combination: 2
Design Moment Profile
DTHA average response
Design Moment Profile
DTHA average response
DDBD Moment Profile
Effective Design Strength
Effective Design Strength
DDBD Moment Profile
12
12
11
11
10
10
9
9
8
8
7
Level
7
Level
6
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1000
2000
3000
4000
Inner column Moments [ KN m ]
(c) Dynamic input :2nd set; Penalty combination:1
0
1000
2000
3000
4000
Inner column Moments[ KN m ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.19 Frame moment profile: comparison between DDBD hypothesis and DTHA average response
96
Chapter 5. Design Verification through Pushover and Nonlinear Time History
For these reason, not more sustained by the wall system, the excess of shear and moment
demand is shifted to the frame columns, and in particular to the inner (core) columns
characterized by an higher stiffness in the transverse direction rather than the perimeter ones.
An accurate study of the action’s time-evolution both in wall and frame columns sustained
and reinforced the previous hypothesis. Considering, for example, the case (a) numerical
model subjected to the EQK1 dynamic record, the development of shear and moment forces
during DTHA will be carefully analysed. To the purpose, seven consecutive instant are taking
into consideration, within the time interval in which the wall flexural strength is completely
mobilized. In this case, the instances considered are t1=8.16 sec, t2= 8.20 sec, t3=8.26 sec,
t4=8.30 sec, t5=8.40 sec, t6=8.60 sec and finally t7= 8.80 sec at which the maximum flexural
strength is reached. In accordance to the set of instants selected, the Figure 5.20 presents the
time-evolution of wall shear, wall moment and frame shear in inner columns. It is easily to
recognize that when the shear and moment action remain in the elastic range for the wall
strength capacity, the inner columns shear profile maintains a fairly constant pattern along the
entire length with contained magnitude values. While advancing the dynamic time-history
analyses, the progressive increment of wall shear and moment demands is accompanied by a
global increment in the shear actions sustained by frame columns. This process linearly
increases, till the maximum wall flexural strength at the seventh instant t7= 8.80 sec is
reached.
t = 8. 20 sec
t = 8. 26 sec
t = 8. 30 sec
t = 8. 40 sec
t = 8. 60 sec
12
12
11
11
11
10
10
10
9
9
9
8
8
8
7
7
7
6
Level
12
Level
Level
t = 8. 16 sec
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
-10000
-5000
0
5000
10000
1500
0
-150000 -100000 -50000
0
50000
100000 150000
Wall Shear [ KN ]
Wall Bending Moment [ KN m ]
(a) Wall Shear
(b) Wall Moment
t = 8. 80 sec
0
-1000
-500
0
500
1000
Frame Shear [ KN ]
(c) Frame shear
Figure 5.20 Wall and frame action during EQK_1 record at the time interval [8.20;8.80]
97
1500
2000
12
12
11
11
11
10
10
10
9
9
9
8
8
8
7
7
7
6
Level
12
Level
L ev e l
Chapter 5. Design Verification through Pushover and Nonlinear Time History
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
-15000
-10000
-5000
0
5000
10000
0
-150000
15000
Wall Shear [ KN ]
t = 9. 80 sec
-100000
-50000
0
50000
100000
0
-2000
150000
t = 9. 86 sec
t = 9. 90 sec
t = 9. 92 sec
t = 9. 94 sec
11
11
11
10
10
10
9
9
9
8
8
8
7
7
7
Level
12
6
6
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
5000
Wall Shear [ KN ]
(a) Wall Shear
10000
15000
0
-150000
-100000
-50000
0
50000
100000
Wall Bending Moment [ KN m ]
(b) Wall Moment
0
500
1000
1500
2000
150000
t = 9. 98 sec
6
5
-5000
-500
t = 9. 96 sec
12
-10000
-1000
Frame Shear [ KN ]
12
0
-15000
-1500
Wall Bending Moment [ KN m ]
Level
Level
6
0
-2000
-1500
-1000
-500
0
500
1000
Frame Shear [ KN ]
(c) Frame shear
Figure 5.21 Wall and frame action during EQK_1 record at the time intervals [8.8 0;9.8] and [9.8;9.98]
98
1500
2000
Chapter 5. Design Verification through Pushover and Nonlinear Time History
At this time, a drastic scatter in column shear profile can be observed isolating completely the
relative shear curve depicted as a viola line in part c of Figure 5.20. Is also evident how the
shear demand is concentrated at the first storey, immediately above the wall plastic hinge
location. Limited, instead, appear the frame actions in the upper storeys where the wall lateral
elastic stiffness is completely maintained.
Once created, the presence of the plastic hinge at the wall base maintains a deep influence on
the global redistribution of actions experienced by the entire lateral resistant system.
Observing successive time intervals, in fact, the frame element at the first storey seems to
keep memory of the maximum shear force experienced (Figure 5.21). In other words, with the
creation of the plastic hinges and the consequent change in the configuration of system’
stiffness properties at the various level, different proportions of seismic actions between
frames and walls take place depending on the particular regions observed. In the regions
undergoing inelastic deformation the change in the proportional factors (βF and βW) appear
evident, while the regions characterized by an elastic deformation field seems conserve the
original design proportion between the allocation of seismic actions.
Capture by SeismoStruct, this phenomenon should assume minor importance in the real case,
where the wall transverse reinforcement operate in order to guarantee an excellent resistance
of the also under plastic conditions. Taking into account only the contribution offered by
longitudinal reinforcement, the computer code adopted SeismoStruct [v 4.09 built 992]
individualizes and perfectly isolates the phenomenon, giving amply confirmations of the
design hypothesis performed. In particular, has been ascertained the a collapse mechanism
characterized by the development of plastic hinges at the wall base, where the shear actions
result most severe. Nevertheless, the numerical results indicate an extension respect to design
previsions of the plastic hinges’ length presumably due to the amplificationinduced by the
higher mode effects both in shear and moment demands.
5.4.8 Longitudinal direction: Wall shear force and moments.
Avoiding tedious repetitions, the results obtain in longitudinal direction are not directly
included in this chapter. An exception has been made for the wall actions, offering the
opportunity of interesting observations.
In particular, considering case (b) and case (d) numerical models, the maximum absolute
values of shear and moment demand are respectively depicted in Figure 5.23 and Figure 5.23.
Following the usual conventions, in each chart is so included the DDBD design prevision and
the capacity envelopes, the first depicted as a green line while the second as a red solid line.
The shear capacity envelope appears able to predict with extremely accuracy the actual shear
demand along the entire height of the building, with a light underestimation at the base level.
Therefore, the capacity design provisions adopted result very well calibrated to quantify the
actual amplification induced by higher mode effects. The huge scatter that separate the
capacity envelope from the DDBD profile appears, now, perfectly justified.
On the other side, the moment capacity envelope results completely unconservative, with a
marked underestimation of moment demand in the higher level of the structure (Figure 5.23).
In fact, the response strength at the base of the wall nearly corresponds to the design strength
at the development of the maximum displacement, which is a direct output of the DDBD
99
Chapter 5. Design Verification through Pushover and Nonlinear Time History
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
procedure. In contrast, higher strength magnitudes are shown at other levels of the wall where
either elastic or moderate levels of inelastic response are expected.
6
6
5
5
4
4
3
3
2
2
1
1
0
-5000
0
5000
10000
0
-5000
Shear [ KN ]
0
5000
10000
Shear [ KN ]
1
(b) Dynamic input : 1st set; Penalty combination: 2
(d) Dynamic input :2nd set; Penalty combination:2
12
12
11
11
10
10
9
9
8
8
7
7
L evel
L evel [ m ]
Figure 5.22 DTHA maximum shear experienced by 4m wall in longitudinal direction
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
10000
20000
30000
40000
Wall Moment [ KN m ]
(b) Dynamic input : 1st set; Penalty combination: 2
0
10000
20000
30000
40000
Wall Moment [ KN m ]
(d) Dynamic input :2nd set; Penalty combination:2
Figure 5.23 DTHA maximum moment experienced by 4m wall during in longitudinal direction
100
Chapter 5. Design Verification through Pushover and Nonlinear Time History
Recalling the observation already advanced in section 2.5.3, the DDBD moment profile and
the overstrength moment envelope are introduced in the following graphs, respectively
represented as a dashed green and as a dashed black curve. Explicit results how the DDBD
moment profile perfectly matches the average response profile and represents the best
prediction of actual response profile. This result confirms that the definition of capacity
moment profile as bilinear envelope, based only on the estimation of the base and mid-height
moments, is not sufficient to provide enough safety against earthquake actions.
12
12
11
11
10
10
9
9
8
8
7
Capacity Envelope
6
DTHA Average Response
DDBD Moment Profile
5
Overstrength Moment profile
Level
Level
7
6
5
4
4
3
3
2
2
1
1
0
-40000 -20000
0
20000 40000
Wall Moment [ KN m ]
0
-40000
-20000
0
20000
40000
Wall Moment [ KN m]
Figure 5.24 Average of maximum moment experienced by inner columns during DTHA
In this sense, is urgently required the analytical definition of new moment capacity envelopes.
The suggestions advanced by Goodsir in 1985, should represent, for example, an important
step for the development of efficacious capacity guidelines. Based on extended studies on
wall and frame seismic behaviour, Goodsir [Goodsir, 1985] recommends a linear moment
envelope that uniformly reduces its magnitude from the base to the top of the structure. In
particular, this moment envelope assumes the value addressed by the elastic moment demand
at the base while at the top coincides with the maximum moment magnitude present upon the
contraflexure height, moment as illustrated on the right-side of Figure 5.25.
A very alternative and more cost-effective recommendation for the capacity design of walls
against the upper storey moments is provide by Sullivan in 2006. In his recent work, Sullivan
suggests to allow the walls to yield over their mid-height demonstrating that, by providing
only limited flexural strength in the walls, excessive curvature ductility demands do not
develop. In other words for bending moment demand, it appears that by accepting some
yielding and detailing the walls accordingly, the structures will be somewhat protected from
101
Chapter 5. Design Verification through Pushover and Nonlinear Time History
changes in intensity. Besides, the wall yielding limits the higher mode actions with a
consequent reduction in the storey drifts at the base and the top of the structures and a general
decrement in the shear demand over the entire wall’s height. Moreover, the consequences of
yielding in the upper levels of a wall are not likely to be excessive detrimental to the global
response of the structure and therefore could be accepted in extreme events.
SHEAR
ENVELOPE
H
50% Vb,wall
Magnified
Design Shear
2H/3
H
MOMENT
ENVELOPE
Magnified
Design Moment
H/3
Eslastic 1st mode
moment demand
Vb,wall = ω v ' Vb,total
V
M
Figure 5.25 Shear and moment capacity design envelopes proposed by Goodsir [1985]
A practical recommendation would therefore be to continue the flexural reinforcement from
the base of the wall up to the mid-height of the structure. Above the mid-height, the
reinforcement could be gradually reduced to a minimum value at the roof level, assuming that
the demand decreased linearly to the top of the building.
5.5 Closing remarks regarding the seismic design verification
Providing the most accurate method for verifying non linear seismic response, inelastic timehistory analyses have been carried out considering both the principal directions. Exploiting
SeismoStruct [v. 4.0.9 built 992] tools, four distinct numerical models are considered in order
to investigate the actual influence in the final result using different dynamic inputs and
different combination of code’s parameter (such as penalty function exponent).
The maximum displacement, drifts, forces and moment developed in frames and walls are,
then, analysed allowing a direct comparison with the deformations and the actions effectively
foreseen in DDBD procedure. In particular, the DTHA results show a perfect agreement with
the DDBD design hypotheses adopted: not only the analytical shape but also the magnitudes
are very well matched.
A source of uncertainties could affect, however, the benignity of the numerical outcomes
obtained. As stated in section 4.2.1, in fact, the particular WCMs analogy adopted to
introduce the three-dimensional configurations of the U-shape walls may provide a level of
flexural strength higher than how intended to the core structures. In particular, the strong
collaboration established between the three singular components initially designed as dinstict
individual members (the flanges and the web), could create a structural system characterized
by a flexural resistance major than that indicated by the design procedure. Beyond the scope of
this work, future studies can improve the reliability and adhesion of core structures elements to the
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Chapter 5. Design Verification through Pushover and Nonlinear Time History
design hypothesis providing not only specific design provisions but also clear and trustworthy
methods to simulate their seismic behaviour numerically. Nevertheless, the excellent match obtained
in the numerical analyses constitutes an incontrovertible proof for the efficacy and truthfulness of
DDBD design techniques based on dual wall-frame structures.
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Chapter 6. Sources of uncertainty associated with research findings
6 SOURCES OF UNCERTAINTY ASSOCIATED WITH
RESEARCH FINDINGS
A rather complete study on 3D response of a dual frame-wall structure has been developed in
the previous chapters. However, the verification procedures have a purely analytical character
and several approximations have been made during the design process and the non-linear
time-history analyses. This section is therefore dedicated to highlight and identify the issues
which might add uncertainty to the findings illustrated thus far.
6.1 ELASTIC VISCOUS DAMPING
Some disagreement exists amongst the scientific/engineering community with regards to the
use of equivalent viscous damping in non linear time-history analysis to represent energy
dissipation sources that are not explicitly included in the model, such as the friction between
structural and non-structural members, friction in opened concrete cracks, energy radiation
through foundation, etc, etc. In fact, some authors [e.g. Wilson, 2001] strongly suggest to
avoid any formulation of the equivalent viscous damping in the numerical models, whilst
others [Priestley and Grant, 2005; Hall, 2006] strongly advice its employment suggesting the
adoption of alternative formulation to the classical damping model.
Proportional to both mass and stiffness, the classical damping models (such as the Rayleigh
damping model) can often be significantly grater than the elastic damping actually active
during the peak seismic response, reducing the effective displacement which undergoes the
analysed structural system. This discrepancy in the damping magnitude is related to the
progressive degradation of the structure’s stiffness during the loading history that leads to
change in the modal period and in the associated vibration frequency. For this reason Priestley
and Grant [2005] suggest to adopt only ductility dependent or stiffness-proportional damping
formulation aiming to guarantee, during the peak seismic response, the achievement of the
most suitable damping value for the structure in exam.
A comparison between the different damping model can, therefore, shows significant scatters
in the actual values of equivalent viscous damping to employ in the dynamic time-history
analysis. An accurate calibration should be, then, performed case per case considering the
material type (e.g. RC, steel, etc), the structural configuration, the deformation level and the
modelling strategy. If fibre modelling approach is adopted, for example, the cracking of the
element is explicitly account and does not need to be represented by means of equivalent
viscous damping, as is done instead in plastic hinge modelling using bilinear momentcurvature relationships.
104
Chapter 6. Sources of uncertainty associated with research findings
As mentioned in Chapter 4, SeismoStruct [v.4.0.9 built 992] is a fibre finite element package
capable of predicting the large displacement behaviour of space structure under static or
dynamic loading, taking into account both geometric nonlinearities and material inelasticity.
Therefore performing the nonlinear dynamic analysis, the dissipation of the majority of
energy introduced by the earthquake action is implicitly included as hysteretic damping
within the nonlinear fibre model formulation of the inelastic frame elements or within the
nonlinear force-displacement response curve formulation used to characterise the response of
link elements. Considering the remaining small quantity of non-hysteretic type of energy
dissipation and the uncertainty related to the different damping formulations, no additional
viscous damping was introduced in the modelling assumptions, conducting conservatively the
verification studies. A sensitivity study on the different elastic viscous damping model should
be, therefore, addressed in future research.
6.2 ROLE OF FLOOR DIAPHRAGMS
Horizontal floors in buildings structures carry most of the gravity loads and so they are
primarily subjected to the inertia forces due to earthquakes. These forces must then be
transferred to lateral resisting systems. Both stiffness and strength of diaphragms are
involved: strength is needed to transfer inertia loads to the lateral restraining elements;
stiffness plays a key role in determining the distribution of the storey force among the lateral
resisting elements as soon as they form a redundant (statically undetermined) system. Usually
in the design process, the floor systems are considered to be capable of providing a strong and
relatively stiff horizontal connection between vertical structural elements. In other words, it is
assumed that the floors of the building are able to transmit inertia forces to and from the frame
and wall elements as rigid diaphragms. This chapter investigates whether such approximation
is reasonably or requires particular consideration during design process.
Since it has been demonstrated the walls tend to control the displacements of the entire framewall structures, any differences in displacement profile between the frame and the wall
elements will be due to deformation of the floor slabs connecting them. The influence of
diaphragm flexibility will be assessed through analysis of case study structures with and
without flexible diaphragms. Non linear time history analysis will be then carried out using
3D models with lateral resisting system that possess the design strengths (evaluated with the
hypothesis of rigid diaphragms).
Using the results already obtained for the prototype structure, the same numerical model will
be analysed with the attention to define in the first case rigid diaphragms while in the second
case flexible one’s. In particular, already presented and deeply investigated in Chapter 4 and
Chapter 5, the SeismoStruct “Link Element” numerical model was selected to conduct this
study. The sensitivity analysis has been demonstrated how increasing the flexural stiffness of
the diaphragm further on had a relatively little effect on the analysis outcomes (with respect to
those already obtained in the previous chapters). On the contrary, reducing the stiffness
magnitude of diaphragm’s constraints produced considerable differences in the analysis
results. Therefore, in the next pages, will be show the result obtained for the extreme limit that
actually realized the condition of flexible diaphragm.
105
Chapter 6. Sources of uncertainty associated with research findings
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
R elative H eight ( hi/H )
R elative H eig h t ( h i/H )
1.0
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.2
0.4
0.6
Lateral Displacement [ m ]
0.8
0.0
1.0
st
(a) Dynamic input :1 set; rigid diaphragms
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.5
0.4
1.0
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.8
(b) Dynamic input :2 set; Penalty combination
1.0
0.6
0.2
0.4
0.6
Lateral Displacement [ m ]
nd
R e lativ e H e ig h t ( h i/H )
R elative H eig ht ( hi/H )
0.6
0.0
0.0
0.2
0.4
0.6
Lateral Displacement [ m ]
0.8
1.0
(c) Dynamic input :1nd set; flexible diaphragms
0.0
0.2
0.4
0.6
Lateral Displacement [ m ]
0.8
1.0
(d) Dynamic input :2nd set; flexible diaphragms
Figure 6.1 DTHA’s wall displacement profiles in transverse direction: comparison between rigid and
flexible diaphragms conditions.
The difference in the analysis outcomes involves not only the displacement shape profiles but
also the magnitude of the displacement effectively experienced by wall and pilastrades. In
Distinguishing rigid and flexible diaphragm’s conditions, for example, in Figure 6.1 and
Figure 6.2 are illustrated the displacements sustained by the 8 m wall during the dynamic
time-history analyses conducted in transverse direction.
106
12
12
11
11
10
10
9
9
8
8
7
7
L e ve l
Level
Chapter 6. Sources of uncertainty associated with research findings
6
6
5
5
4
4
3
3
2
2
1
1
0
0.00
0.20
DDBD Displacement
0.40
0.60
0
0.00
0.80
DTHA Displacement Response
Displacement [ m ]
0.40
0.60
0.80
DTHA Displacement Response
Displacement [ m ]
(a) Dynamic input :1st set; rigid diaphragms
(b) Dynamic input :2nd set; rigid diaphragms
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
0.20
DDBD Displacement
6
5
6
5
4
4
3
3
2
2
1
1
0
0.00
0.20
Design Displacement
0.40
0.60
0.80
Average DTHA Displacement
Displacement [ m ]
(c) Dynamic input :1nd set; flexible diaphragms
0
0.00
0.20
Design Displacement
0.40
0.60
0.80
Average DTHA Displacement
Displacement [ m ]
(d) Dynamic input :2nd set; flexible diaphragms
Figure 6.2 DTHA’s wall displacement profiles in transverse direction: comparison between rigid and
flexible diaphragms conditions.
In particular, in Figure 6.1 depicts the maximum displacements experienced by the 8 m wall
in each DTHA, while in Figure 6.2 the DDBD design displacement is directly compared with
the average of the maximum DTHA displacement profiles. Appears evident how imposing the
flexible diaphragm’s condition leads to a complete disconnection between the two lateral
resistant systems: the wall and frame system.
107
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
R e la tiv e H e ig h t ( h i/H )
R e la tiv e H e ig h t ( h i/H )
Chapter 6. Sources of uncertainty associated with research findings
0.6
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.2
0.4
0.6
0.8
Lateral Displacement [ m ]
1.0
1.2
(a) 8m wall displacemnt;
Dynamic input :1st set; flexible diaphragms.
0.0
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.2
0.4
0.6
0.8
Lateral Displacement [ m ]
1.0
1.2
(b) 8m wall displacemnt;
Dynamic input :2st set; flexible diaphragms.
R elative H eight ( hi/H )
R elative H eig ht ( hi/H )
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Lateral Displacement [ m ]
1.2
(c) Inner Frame outer column displacement;
Dynamic input :1st set; flexible diaphragms.
0.0
0.2
0.4
0.6
0.8
Lateral Displacement [ m ]
1.0
1.2
(d) Inner Frame, outer column displacement;
Dynamic input :2nd set; flexible diaphragms.
Figure 6.3 Displacement profiles assumed under flexible diaphragm conditions by wall and inner
pilastrades during DTHA in transverse direction.
108
12
12
11
11
10
10
9
9
8
8
7
7
Level
L e vel
Chapter 6. Sources of uncertainty associated with research findings
6
6
5
5
4
4
3
3
2
2
1
1
0
0.00
0.20
0.40
Design Displacement
0.60
0.80
1.00
0
0.00
1.20
0.20
11
11
10
10
9
9
8
8
7
7
Level
L evel
12
6
5
4
4
3
3
2
2
1
1
Design Displacement
0.60
0.80
1.00
1.20
6
5
0.40
0.80
Average DTHA Displacement
(b) 8m wall displacemnt;
Dynamic input :2st set; no rigid diaphragm.
12
0.20
0.60
Displacement [ m ]
Displacement [ m ]
(a) (8m wall displacemnt;
Dynamic input :1st set; no rigid diaphragm.
0
0.00
0.40
Design Displacement
Average DTHA Displacement
1.00
1.20
Average DTHA Displacement
0
0.00
0.20
0.40
Design Displacement
0.60
0.80
1.00
1.20
Average DTHA Displacement
Displacement [ m ]
Displacement [ m ]
(c) Inner Frame outer column displacement;
Dynamic input :1st set; no rigid diaphragm.
(d) Inner Frame, outer column displacement;
Dynamic input :2nd set; no rigid diaphragm.
Figure 6.4 Displacement profiles assumed under flexible diaphragm conditions by wall and inner
pilastrades during DTHA in transverse direction.
109
Chapter 6. Sources of uncertainty associated with research findings
Huge scatter distinguishes the wall and frame displacement profiles especially at the lower
storeys. This difference become more significant increasing the distance between the wall and
frames, reaching its maximum for the central inner frame. the Both the lateral resistant system
return to assume their characteristic shape profiles: the peculiar convex shape for wall
elements while the typical concave shape indicates the frame elements (Figure 6.3 and Figure
6.4). The importance of these results is amplified considering the initial geometric
configuration: the structural layout is simple, regular and compact organized around a twoway warping distributed along the entire floor surface. This structural configuration
undoubtedly facilitates the distribution of displacements and actions fields to the overall
system but its efficacy is not able, however, to assure or reproduce the rigid diaphragm
constraint conditions.
The characterization of the actual strength and stiffness slab properties appears, therefore, of
primary importance for a correct restitution of a reliable seismic response. In fact, not only an
alteration in displacement magnitudes and shape profiles can be observed varying the floor
stiffness, but also a significantly different proportion of the base shear between the wall and
frame systems can be argued. Moreover the incorporation of flexible diaphragm within the
models tends to lengthen the period vibration, as Table 6.1 testifies. Typically, if the period of
vibration of a structure increases, an increment in the displacements field, which undergoes
during an earthquake attack, should be expected.
Table 6.1 Initial periods of 3D models with and without flexible diaphragms.
1
2
3
4
Vibrational Mode
Rigid diaphragm
model
Flexible diaphragm
model
Percentice
Differences
[-]
Period T [s]
Period T [s]
[%]
1ST Vibrational Mode in x-direction
1.345
1.710
21%
ST
Vibrational Mode in y-direction
1.118
2.243
50.1%
ND
Vibrational Mode in x-direction
0.282
1.452
80.6%
2ND Vibrational Mode in y-direction
0.211
1.476
85.7%
1
2
RD
Vibrational Mode in x-direction
0.107
1.028
89.6%
RD
Vibrational Mode in y-direction
0.107
0.809
86.7%
3
3
On the other hand, in common practice should be emphasized the importance of floor stiffness
design and verification. In fact, in general, the diaphragms are checked for strength, on a
separated model, while no check is generally carried out concerning stiffness. This lack of
attention can explain experimental evidences: past earthquakes have shown many structures
suffering for problems connected to floors flexibility, since it was not taken correctly into
account during design. Therefore, the seismic behaviours have appeared so different from the
design hypothesis and many structural problems arise due to the determination of erroneous
collapse mechanisms. This should induce crucial reflections on the designer’s mind: when is
the rigid diaphragm assumption acceptable? How much should be thrust to create stiff
diaphragms, in order to make the structure similar to our model? Interrogative of not easy
solution, these questions involves many different aspects: from the design hypothesis, to the
modelling characterizations and, not least, to the effective constructive chooses. Beyond the
110
Chapter 6. Sources of uncertainty associated with research findings
scope of this work, should be auspicial that future works will face this problem promoting not
only design provision but also realistic building solutions.
6.3 U-SECTION MODELLING UNCERTAINTIES
Utilized to model the U-shape walls present in the prototype structure, the “wide-column
analogy” (known also as the “equivalent frame method”) represents one of first modelling
approach to simulate the core structure behaviour. Despite the frequent use of wide-column
models (WCMs) in engineering practise, literature on wide-column models is scarce with
exception of the recent work conducted by Beyer et al. [2008] on torsionally eccentric
buildings with U-shape RC walls. Therefore, this section will be principally referred on the
analysis’ results obtained in that study.
As illustrated in 4, in WCMs of non-planar walls the web and the flange sections are
represented by vertical column elements located at the centroid of the web and flange
sections. These vertical elements are then connected by horizontal links running along the
weak axis of the sections having common nodes at the corners. Except for a torsional
flexibility, which can be assigned, these links are modelled as rigid.
Representing a simple, efficient and computational inexpensive approach, the WCMs of
structural walls are often preferred to models with 2D or 3D, despite they represent only an
approximations of the real structural system and also exhibit some drawbacks. Three are the
main drawbacks of WCMs: the parasitic bending moments due to shear stresses along the
wall edges, the response to torsional loading and the rotation demand on coupling beams. The
latter does not concern the U-shape walls that are completely open on one site, as our
prototype structure and its discussion is therefore leaved out.
A wide-column module for wall section comprises one flexible column element and two rigid
links which extend over the width of the wall. Stafford-Smith and Girgis [1986] found that
such elements are afflicted by parasitic bending moments when continuous shear stresses
along the wall edges are modelled. Since continuous distributed shear stresses along a wall
edge will be lumped into discrete shear forces at the rigid links these shear forces will cause
reverse bending of the column element. Even if not directly adopted in numerical model of
the prototype structure (in order to avoid excessive computational burden), considerable
efforts by a number of researchers have been undertaken to improve the wide-column model
and to overcome its shortcomings related to the parasitic moments (e.g. Kwan [1993]; Kwan
[1994]; MacLeod and Hosny [1977]; Stafford-Smith and Abate [1981]; Stafford-Smith and
Girgis [1984]; Stafford-Smith and Girgis [1986]).
Regarding the second drawback, the core structures which are partially closed by deep beams
may lead to erroneous results for torsional loading since a large portion of the torsional
resistance is attributed to Saint-Venant shear stresses rather than warping of the section
[Stafford-Smith and Girgis [1986]. The Saint-Venant shear stresses cause parasitic bending
moments, which lead to reverse bending of the web and flanges. As a consequence the
behaviour of the WCM might be dominated by the reverse bending rather than deformations
that account for physical-meaningful stresses. However the torsional behaviour of U-shape
walls and related modelling aspects is the object of present research projects and a more indepth discussion will be therefore untimely.
111
Chapter 6. Sources of uncertainty associated with research findings
A part from the drawbacks, many other modelling assumptions and parameters deeply
influence the properties and the performance of WCMs structures under inelastic seismic
behaviour. They can be summarized into five main arguments (Beyer [2008]):
-
Subdivision of the U-shape section into planar wall section;
-
Vertical spacing of the horizontal links;
-
Properties of the links;
-
Number of vertical elements representing the planar wall sections between horizontal
links (a wall section between two horizontal links is typically subdivided into a number
of column elements);
-
Properties of the vertical elements such as the axial, flexural, shear and torsional
stiffnesses;
For the sake of completion, each argument is briefly discussed and compared with the model
assumption performed in this study.
(a)
Subdivision of non-planar wall sections.
The wide column analogy requires the subdivision of non-planar wall sections into planar
subsections, which can typically be undertaken in different ways. Figure 6.5 shows three
possible division schemes for U-shaped wall that differ only regarding the allocation of
the corner areas to the web and flange sections.3
lF
Scheme A
lW
lF
lW
lW
Scheme B
Scheme C
Figure 6.5 Different schemes for subdividing the U-shaped section into planar wall section [Beyer et al.,
2008]
The differences on the gross section properties, in the moments of inertia about the x-and
y-axes are so small to be considered hardly of significance in seismic engineering, in
particular if the structure is expected to respond in the inelastic range. However
performing some sensibility analysis, the scheme C led in most cases to moments and
forces that were between those of the other two models (Beyer et al. [2008]). Hence, this
subdivision schemes seems suitable if the WCMs is analysed for different directions of
loading and adopting in the prototype structure numerical models. Following this scheme,
the corner area is split between the flange and the web elements and the links join where
the web and flange elements meet.
112
Chapter 6. Sources of uncertainty associated with research findings
(b)
Spacing of horizontal links.
The spacing of the horizontal links affects the apparent shear stiffness and the magnitude
of the parasitic bending moments which occur as a consequence of the transmission of
shear forces from the link spacing to the wall elements. Stafford-Smith and Girgis [1986]
suggest to limit the link spacing to one-fifth of the overall wall height in order to control
the deformations due to parasitic bending moment. However adhering to the engineering
common practice when the U-shaped wall is modelled as part of an entire building, in the
prototype structure this threshold has been further reduced spacing the link in relation to
the storey’s heights even if this assumption might lead to a softer structure than intended.
(c)
Properties of the horizontal links.
As suggested by Reynouard and Fardis [2001], in the prototype structure the links has
characterized by infinite flexure and shear stiffnesses. In order to avoid the rise of some
numerical errors in SeismoStruct [v. 4.0.9 built 992] models, also the torsional stiffness is
assigned as infinite. However, with the development of more refined versions of the
software code and the increment in the computational computer’s power, it will be hoped
to use the elastic torsional stiffness of the links as Reynouard and Fardis [2001] suggested.
In her recent work Beyer (Beyer et al. [2008]) proposed to also to assign an in-plan shear
flexibility to the links. This shear flexibility should be introduced to account for
deformations in the physical experimental test, which were caused by vertical shear
stresses transmitted from the web to one flange along the corners.
(d)
Number of the vertical element between links.
The option of connect two consecutive horizontal links with a single vertical element
would be the simplest solution. However, in regions undergoing significant inelastic
deformations, this modelling assumption will lead to not accurate analysis outcomes
especially if displacement–based (DB) finite element formulation is adopted by software
code. In fact, cubic Hermitian polynimials are usually used as displacement shape
functions in the DB element formulation, corresponding for instance to a linear variation
of curvature along the entire element’s length. Therefore a refined discretization of the
structural element (typically 2 or 3 elements per structural member) is required in order to
capture large nonlinear variation.
In the prototype structure numerical models, however, the adoption of the link spacing
equal to the storey height limits this problem and was assumed that only one element was
sufficient to connecting two consecutive horizontal links.
(e)
Properties of the vertical elements such as the axial, flexural, shear and
torsional stiffnesses.
The shear and torsional stiffnesses of a web of flange section are dependent on the state of
cracking and the axial and flexural strains. These quantities vary during the course of the
loading but in most structural analysis programs shear, flexural and torsional stiffnesses
are considered as constant values during the entire loading history. Moreover their
magnitudes are automatically computed by the code starting from the geometry and the
material properties of each element. This observation leads to differentiate the section’s
properties for the elements which undergoing inelastic deformations and the elements that
remain largely uncracked and elastic. In other word, if the U-shape wall is a part of a
larger structure, the distribution of the shear, flexural and torsional stiffness should reflect
the distribution of the expected inelastic deformation (Beyer et al. [2008]).
113
Chapter 6. Sources of uncertainty associated with research findings
Most of the previous observations have been taken into account during the definition of the
prototype structure’s numerical models. However, some simplifications have been introduced
in the modelling approach of U-shape structure to not incur in computational errors or vain
numerical efforts. In fact, not the most refined characterization of the core structure but the
global verification of dual frame-wall structure’ seismic response was the primer object of the
research project. Therefore the development of a simple, efficient and computational
inexpensive analysis models has been assumed a character of primary importance for the
entire study, where inelastic analysis with complex displacement or acceleration field are
applied to a large three-dimensional model (thirteen level, 768 m2 per floor). However, the
possible refinement or implementation of the numerical models adopted could represent an
interesting theme for future research studies.
114
Chapter 7. Conclusions
7 CONCLUSIONS
Considering two different set of artificial ground motion records, several dynamic timehistories analyses are carried out in order to verify the efficiency and consistency of DDBD
design techniques focused on dual frame-wall structure. For this propose, a prototype
structure consists of two-way moment-resisting structural frames with channel walls of
reinforced concrete at each end of the building is examined. Starting from the firsts steps of
the procedure, a complete design of the building was performed considering the flexural
strength requirements indicate by DDBD design method. Important hints emerge for a further
implementation of capacity design guidelines, especially regards wall bending moment
envelope.
7.1 Displacement–Based Designed method for dual frame-wall structure
Exploiting the symmetric properties characterizing the structural layout, nonlinear dynamic
time-histories analysis have been performed in the two principal directions, the transverse and
longitudinal one. Even if the percentage discrepancies between DTHA results and DDBD
response quantities tend to increase in longitudinal direction, the same seismic response can
be observed in both the cases. Therefore, avoiding tedious repetitions, only the results
obtained for transverse direction has been presented and commented in detail. The DTHA
results show a perfect agreement with the DDBD design hypotheses adopted: not only the
analytical shape but also the magnitudes are very well matched. In particular, summarizing
the most important aspects highlighted during the analyses phase, can be observed that:
-
An excellent match between the DDBD displacement profile and the DTHA average
displacement profile characterizes the performance of the prototype structure
subjected to earthquake loadings. The only exception regards the first floor, where an
overestimation lightly affects the design displacement profile.
-
The shape assumed by the average storey drift profile is very closed to that proposed
by DDBD design techniques, but higher values than expected are recorded for the
intermediate levels where also the code drift limit is exceeded.
-
The DTHA moment and shear response profiles for frame columns are fairly constant
as DDBD procedure suggests. Only two the notable non-linearity: at the top and
bottom of the building. Particular importance assumes the discrepancy noticed at the
first storey revealing the close interaction between frame and wall during the
development of plastic hinges zones.
Should be, however, mentioned the light underestimation of shear magnitude that
characterizes all the analysis case examined.
115
Chapter 7. Conclusions
-
The wall shears profile close matches the capacity design envelopes, even if a more
conservative estimation of base shear should be appreciable. On the other side the
capacity design envelope for wall bending moment is not sufficient to guarantee
enough safety against higher modes effects. Noticed in the transverse direction, this
observation becomes evident for the 4m longitudinal walls, where the capacity criteria
adopted do not allow neither the complete recovering of DDBD moment profile.
Besides, interesting results are also obtained performing the sensitivity analyses focused both
on modelling aspects and dynamic input combinations:
-
In SeismoStruct [v. 4.0.9 built 992] can be activated special constraint elements’ tools.
to simulate the presence of rigid floor diaphragm or pinned connection Even if
extremely ductile, their use needs an accurate calibration of some code’s parameters
such as the penalty function coefficients. In fact, only a correct setting guarantees
stable and reliable results for all type of analyses.
-
Obtained as the average of the maximum response quantities recorded during seven
different time-histories, the reliability and stability of DTHA results are verified using
two different set of dynamic input motions. Even if the peculiar characteristics of
artificial earthquake (especially in term of frequency contents) can strongly influence
the global response of the structure, the final response appears to be not so sensitive to
the particular set analysed. Therefore, it was proved that the performance of seven
different DTHAs is sufficient to guarantee stable and reliable results. However, with
the modern computer power seems logical to increase the global number of records
above the minimum of seven, to ensure a more representative average [Priestley et al.,
2007].
7.2 Hints for new capacity design guidelines for frame-wall structure
In section 5.4.8 has been demonstrate how the definition of wall moment capacity profile as
bilinear envelope is not sufficient to provide enough safety against earthquake actions if based
only on the estimation of the base and mid-height moments. In this sense, is urgently required
the analytical definition of new moment capacity envelopes.
The suggestions advanced by Goodsir in 1985, should represent, for example, an
important step for the development of efficacious capacity guidelines. Based on extended
studies on wall and frame seismic behaviour, Goodsir [Goodsir, 1985] recommends a linear
moment envelope that uniformly reduces its magnitude from the base to the top of the
structure. In particular, this moment envelope assumes the value addressed by the elastic
moment demand at the base while at the top coincides with the maximum moment magnitude
present upon the contraflexure height. This new moment envelope should represent a very
good agreement between the actual wall’s seismic response and the maintenance of a simple
linear shape.
Otherwise, a very alternative and more cost-effective recommendation for the capacity
design of is provide by Sullivan in 2006. In his recent work, Sullivan suggests to allow the
walls to yield over their mid-height demonstrating that, by providing only limited flexural
strength in the walls, excessive curvature ductility demands do not develop. A practical
recommendation would therefore be to continue the flexural reinforcement from the base of
the wall up to the mid-height of the structure. Above the mid-height, the reinforcement could
116
Chapter 7. Conclusions
be gradually reduced to a minimum value at the roof level, assuming that the demand
decreased linearly to the top of the building.
Finally, the definition of a reliable capacity design can not leave out the complex phenomenon
arising during the development of plastic hinge at the wall base level. In fact, once the fully
flexural and shear strength capacity are reached in the wall, a re-distribution of the exceeding
solicitations takes place among the other lateral resistant elements, depending on their relative
stiffness proportion. Consequently, a drastic increment of actions experienced by the columns
can be observed, especially at those levels in the immediate proximity of plastic hinge
location (in this case, between the ground floor and the first storey). Therefore is absolutely
need a more accurate prediction of the seismic forces effectively acting above the plastic
hinge zones as far as a detailed design of plastic hinge locations itself.
7.3 Future research
Future research on seismic design of frame-wall structures should address the issues not
directly included with in the scope of this research. Three, in particular, the relevant aspects.
Appropriate calibration of DDBD procedure for steel frames with RC walls: The design
process has been highlighted the scarcity or complete absence of appropriate
formulations specific for steel structures. The majority of numerical and experimental
studies already performed are relative to RC structures, commonly design with
rectangular or square element sections both for beam, columns and wall members.
Consequently no calibration of the DDBD equations is available for steel section
members usually characterized by a very vast section variety.
In particular, moving from rolled I-shape sections to box or hallow sections, more
detailed capacity design specifications should be provided regard biaxial attack, taking
into account the different properties offered by the section depending on the particular
orientations considered (strong vs weak axis).
Seismic behaviour of irregular structures: The present study has examined a 3D structure
characterized by a regular layout. Considering that the building twist can affect the
proportion of lateral load carried by the frames and walls respectively, the verification of
DDBD design procedure for irregular structural layout should be a task of future
research.
Effect of non-structural components on the torsional response of buildings: Some nonstructural components (e.g. masonry infill panels) are particularly stiff but have only
small displacement capacities. An analytical study could investigate the effect of failure
of non-structural components on the global response of a building. This aspect assumes
a relevant importance if related with the normal distribution of non-structural
components in the plan layout. Following only functional and distributive guide lines,
the positions occupies by infill panels is usually characterized by strong irregularity
Therefore, a marked increment in the accidental eccentricities is expected with a
consequent promotion of torsional twist.
The evaluation of DDBD design procedure and seismic response of three-dimensional framewall structures was based only on the numerical results obtained for the case study structure.
Validating against one project research can represent only a start but not a final conclusion.
117
Chapter 7. Conclusions
It’s essential, therefore, that more experimental results on the cyclic behaviour of dual wallframe structures will become available.
118
References
REFERENCES
Ballio, G., Bernuzzi,C. [2006], Progettare costruzioni in acciaio, HOEPLI,Milano,Italy
Bayer, K., Dazio, A. and Priestley, M.J.N. [2008] Seismic design of torsionally eccentric buildings
with U-shape RC walls, IUSS Press, Pavia, Italy.
Comité Européen de Normalisation, Eurocode 8, Design of structures for earthquake resistance – Part
1:General rules, seismic actions and rules for buildings, prEN 1998-1, December 2004 draft,
Belgium.
Computers and Structures [2005] SAP2000: Linear and Nonlinear Static and Dynamic Analysis and
Design of Three Dimensional Structures, Berkeley, California, USA.
Cosenza, E., Magliuolo, G., Pecce, M., Damasco, R. [2004] Progetto Antisismico di Edifici in
Cemento Armato, IUSS Press, Pavia, Italy.
Dow Nakaki,S., Stanton, J.F.,Srithan,S. [1999], “An Overview of the PRESS Five-Storey Precast
Taset Building,” PCI Journal, pp.26-39
Hall J.F. [2006] "Problems encountered from the use (or misuse) of Rayleigh damping,"
Earthquake Engineering and Structural Dynamics, Vol. 35, No. 5, pp. 525-545.
Kwan, A. K. H. [1993] “Improved wide-column-frame analogy for shear/core wall analysis”, ASCE
Journal of the structural Engineering,Vol.119, No.2, pp.420-437.
Kwan, A. K. H. [1994] “Unification of exixting frame analogies for coupled shear/core wall analysis,”
Computers &Structures, Vol.51, No.4, pp.393-401.
MacLeod, I.A. and Hosny, H. [1977] “Frame analysis of shear wall cores,” Journal of the Structural
Division, ASCE, Vol.103, No.ST10, pp.2037-2047.
Mazzolani, F. M.,.Landolfo, R, Della Corte, G., Faggiano, B. [2006], Edifici con Struttura in
Acciaio in Zona Sismica, IUSS Press, Pavia, Italy.
116
References
Ordinanza del Presidente del Consiglio dei Ministri n. 3274 – 03/03/2003 as modified by Ordinanza
del Presidente del Consiglio dei Ministri n. 3431 – 03/05/2005, Norme tecniche per il progetto, la
valutazione e l’adeguamento sismico degli edifici, Italy.
Pauley, T. [2002], “ A Displacement-focused Seismic Design of Mixed Building Systems,”
Earthquake Spectra, V.18(4), pp.689-718.
Petrini, L., Pinho, R., Calvi, G.M. [2004] Criteri di progettazione antisismica degli edifici, IUSS
Press, Pavia, Italy.
Priestley, M.J.N, Calvi, G.M., Kowalsky, M.J. [2007] Displacement-based Seismic Design of
Structures, IUSS Press, Pavia, Italy.
Priestley M.J.N., Grant D.N. [2005] "Viscous damping in seismic design and analysis," Journal of
Earthquake Engineering, Vol. 9, Special Issue 1, pp. 229-255.
Reynouard, J.M. and Rardis, M.N.[2001] Shear wall structures, No.5 in CAFEELECOEST/ICONS Thematic Report, Laboratorio Nacional de Engenharia Civil (LNEC),
Lisboa,Portugal.
SeismoSoft [2007] SeismoStruct: A computer program for static and dynamic nonlinear analysis of
framed structures (online), available from URL: http://www.seismosoft.com.
Stafford-Smith, B. and Abate, A. [1981] “Analysis of non-planar shear wall assemblies by analogous
frame,” Proceedings of Institution of Civil Engineers, Vol. 71, No. 2, pp.395-406.
Stafford-Smith, B. and Girgis, A. [1984] “Simple analogous frames for shear wall analysis,” Journal
of Structural Engineering, ASCE, Vol.110, No.11, pp2655-2666.
Stafford-Smith, B. and Girgis, A. [1984] “Deficiencies in the wide column analogy for shearwall
analysis,” Concrete International, pp. 58-61
T.J.Sullivan, M.J.N.Priestley, G.M.Calvi [2007], Seismic design of Frame-Wall Structures, IUSS
Press, Pavia, Italy.
Wilson E. [2001], Static and Dynamic Analysis of Structures, Computers and Structures Inc,
Berkeley,
California.
(excerpts
available
at
URL:
www.csiberkeley.com/support_technical_papers.html)
Xenidis,H. and Avramidis, I [1999] ”Comparative performance of code prescribed analysis methods
for R/C buildings with shear wall cores,” Proceedings of the 4th European conference on Structural
Dynamics: EURODIN’ 99, pp.869-875, Blkema, Rotterdam.
117
Appendix A
APPENDIX A
118
Appendix A
A.1 Drift limit at contraflexure height θCF
The design drift θd,lim should be selected as the minimum between the code drift limit θC and
that associated with the ductility capacity curvature of the walls θWall. In particular, θWall can
be approximated to the wall drift limit at contraflexure height θCF, evaluated as:
θ CF =
φ y ,Wall ⋅ H CF
2
+ (φ dc − φ y ,Wall ) ⋅ L p = 0.0244
(0.1)
Where:
φy,Wall is the wall limit- state curvature sets equal to φy,Wall = 0.9x 0.072/ lWall = 0.0081 [m-1];
LP
LSP
k
is the plastic hinge length at the base of the wall, LP=k HCF+ 0.1 lWall+LSP =2.58 [ m ];
is the strength penetration length evaluated as LSP = 0.22 fye dbl = 242
[mm];
is an reduction factor equal to k =0.2 (fu/fy-1) =0.07;
A.2
Column Flexural Design in Transverse Direction
Height
Beam Flexural
Design
External
Column Moment
Internal Column
Moment
External
Column Moment
Design
Internal Column
Moment Design
Hi
Mbeam,1
MC1,f
MC2,f
MC1,des
MC2,des
[ - ]
[ m ]
[ KN m ]
[ KN m ]
[ KN m ]
[ KN m ]
[ KN m ]
12
11
10
9
8
7
6
5
4
3
2
1
0
39.2
36.0
32.8
29.6
26.4
23.2
20.0
16.8
13.6
10.4
7.2
4.0
0
472.9
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
945.8
0.0
668.78
668.78
668.78
668.78
668.78
668.78
668.78
668.78
668.78
668.78
668.78
668.78
709.35
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1337.6
1418.7
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
1043.30
709.35
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
2086.6
1418.7
External
Column Shear
Design
Internal Column
Shear Design
Level
A.3
Column Shear Design in Transverse Direction
External
Column shear
Internal Column
Shear
Storey Frame
shear
Design
Level
Frame Storey
shear
Vs,i
VS,i_EXT
VS,i_INT
Vs,i
VC1,des
VC2,des
[ - ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
12
11
10
9
8
7
6
5
4
3
2
1
0
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
1773.4
0.0
295.6
295.6
295.6
295.6
295.6
295.6
295.6
295.6
295.6
295.6
295.6
295.6
0.0
591.1
591.1
591.1
591.1
591.1
591.1
591.1
591.1
591.1
591.1
591.1
591.1
0.0
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
2766.5
0.0
461.08
461.08
461.08
461.08
461.08
461.08
461.08
461.08
461.08
461.08
461.08
461.08
0.00
922.16
922.16
922.16
922.16
922.16
922.16
922.16
922.16
922.16
922.16
922.16
922.16
0.00
119
Appendix A
A.4
Column Flexural Design in Longitudinal Direction: External Frames (1 and 4)
External
Column
Moment
External
Column
Moment
Design
Internal
Column
Moment
Design
MC1,f
MC1,des
MC2,des
[ KN ]
[ KN m ]
[ KN m ]
[ KN m ]
110.817
199.4
626.87
977.92
1955.8
221.634
358.9
626.87
977.92
1955.8
Height
Beam
Flexural
Design
Storey Shear
Hi
Mbeam,1
Vbeam,1
V
[ - ]
[ m ]
[ KN m ]
[ KN ]
12
39.2
443.3
11
36.0
886.5
10
32.8
886.5
221.634
358.9
626.87
977.92
1955.8
9
29.6
886.5
221.634
358.9
626.87
977.92
1955.8
8
26.4
886.5
221.634
358.9
626.87
977.92
1955.8
7
23.2
886.5
221.634
358.9
626.87
977.92
1955.8
6
20.0
886.5
221.634
358.9
626.87
977.92
1955.8
5
16.8
886.5
221.634
358.9
626.87
977.92
1955.8
4
13.6
886.5
221.634
358.9
626.87
977.92
1955.8
3
10.4
886.5
221.634
358.9
626.87
977.92
1955.8
2
7.2
886.5
221.634
358.9
626.87
977.92
1955.8
1
4.0
886.5
221.634
368.2
626.87
977.92
1955.8
0
0.0
0.0
0.000
664.90
664.90
1329.8
Level
A.5
Level
Beam Shear
Design
0
B,max
Column Shear Design in Longitudinal Direction:External Frames (1 and 4)
Storey
Frame
shear
Vs,i
External
Column
shear
Design
Internal
Column
Shear
Design
External
Column
Moment
Design
Design shear
Frame
External
Column
Shear Design
Internal Column
Shear Design
VS,i_INT
[ KN ]
MC1,des
Vs,i
VC1,des
VC2,des
[ KN m ]
[ KN ]
[ KN ]
[ KN ]
[ - ]
[ KN ]
VS,i_EXT
[ KN ]
12
2216.3
277.0
554.1
977.92
3457
432.19
864.37
11
2216.3
277.0
554.1
977.92
3457
432.19
864.37
10
2216.3
277.0
554.1
977.92
3457
432.19
864.37
9
2216.3
277.0
554.1
977.92
3457
432.19
864.37
8
2216.3
277.0
554.1
977.92
3457
432.19
864.37
7
2216.3
277.0
554.1
977.92
3457
432.19
864.37
6
2216.3
277.0
554.1
977.92
3457
432.19
864.37
5
2216.3
277.0
554.1
977.92
3457
432.19
864.37
4
2216.3
277.0
554.1
977.92
3457
432.19
864.37
3
2216.3
277.0
554.1
977.92
3457
432.19
864.37
2
2216.3
277.0
554.1
977.92
3457
432.19
864.37
1
2216.3
277.0
554.1
977.92
3457
432.19
864.37
0
2216.3
277.0
554.1
664.90
3457
120
Appendix A
A.6
Column Flexural Design in Longitudinal Direction: Internal Frames (2 and 3)
External
Column
Moment
External
Column
Moment
Design
Internal
Column
Moment
Design
MC1,f
MC1,des
MC2,des
[ KN ]
[ KN m ]
[ KN m ]
[ KN m ]
265.8
626.87
977.92
977.92
221.634
451.9
626.87
977.92
977.92
221.634
451.9
626.87
977.92
977.92
886.5
221.634
451.9
626.87
977.92
977.92
26.4
886.5
221.634
451.9
626.87
977.92
977.92
23.2
886.5
221.634
451.9
626.87
977.92
977.92
6
20.0
886.5
221.634
451.9
626.87
977.92
977.92
5
16.8
886.5
221.634
451.9
626.87
977.92
977.92
4
13.6
886.5
221.634
451.9
626.87
977.92
977.92
3
10.4
886.5
221.634
451.9
626.87
977.92
977.92
2
7.2
886.5
221.634
451.9
626.87
977.92
977.92
1
4.0
886.5
221.634
470.5
626.87
977.92
977.92
0
0.0
0.0
0.000
1329.80
1329.80
1329.80
Height
Beam
Flexural
Design
Storey Shear
Hi
Mbeam,1
Vbeam,1
V
[ - ]
[ m ]
[ KN m ]
[ KN ]
12
39.2
443.3
110.817
11
36.0
886.5
10
32.8
886.5
9
29.6
8
7
Level
A.7
Level
Beam Shear
Design
0
B,max
Column Shear Design in Longitudinal Direction: Internal Frames (2 and 3)
Storey
Frame
shear
External
Column
shear Design
Internal
Column
Shear Design
External
Column
Moment Design
Internal
Column
Moment
Design
External
Column Shear
Design
Internal
Column Shear
Design
Vs,i
VS,i_INT
[ KN ]
MC1,des
MC2,des
VC1,des
VC2,des
[ - ]
[ KN ]
VS,i_EXT
[ KN ]
554.1
554.1
[ KN m ]
977.9
[ KN ]
1662.3
[ KN m ]
977.92
[ KN ]
12
995.72
995.72
11
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
10
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
9
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
8
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
7
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
6
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
5
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
4
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
3
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
2
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
1
1662.3
554.1
554.1
977.92
977.9
995.72
995.72
0
1662.3
554.1
554.1
1329.80
1329.8
1329.80
1329.80
121
Appendix B
APPENDIX B
122
Appendix B
B.1
Eigenvalue results for Link Element SeismoStruct model using Lagrange
Multiplier constraint algorithm.
Activating all global mass direction (X,Y,Z,RX,RY and RZ), the following results are
obtained:
SEISMOSTRUCT
Individual Modal Mass
Cumulative Modal Mass
Mode
Period
Ux
Uy
Uz
Ux
Uy
Uz
1
1.346
67.09%
0.00%
0.00%
67.09%
0.00%
0.00%
2
1.119
0.00%
65.94%
0.00%
67.09%
65.94%
0.00%
3
0.883
0.00%
0.00%
0.00%
67.09%
65.94%
0.00%
4
0.283
17.62%
0.00%
0.00%
84.71%
65.94%
0.00%
5
0.239
0.05%
0.00%
0.00%
84.76%
65.94%
0.00%
6
0.239
0.00%
0.00%
1.58%
84.76%
65.94%
1.58%
7
0.239
0.00%
0.01%
0.00%
84.76%
65.95%
1.58%
8
0.239
0.00%
0.00%
0.00%
84.76%
65.95%
1.58%
9
0.226
0.02%
0.00%
0.00%
84.78%
65.95%
1.58%
10
0.229
0.00%
0.00%
0.00%
84.78%
65.95%
1.58%
11
0.229
0.00%
0.00%
0.00%
84.78%
65.95%
1.58%
12
0.226
0.00%
0.05%
0.00%
84.78%
66.00%
1.58%
13
0.226
0.00%
0.00%
0.00%
84.78%
66.00%
1.58%
14
0.229
0.00%
0.00%
1.12%
84.78%
66.00%
2.70%
15
0.229
0.00%
0.00%
0.00%
84.78%
66.00%
2.70%
16
0.226
0.00%
0.00%
5.86%
84.78%
66.00%
8.56%
17
0.219
0.00%
0.00%
0.28%
84.78%
66.00%
8.84%
18
0.219
0.00%
0.06%
0.00%
84.78%
66.06%
8.84%
19
0.212
0.00%
18.57%
0.00%
84.78%
84.63%
8.84%
20
0.205
0.00%
0.00%
51.63%
84.78%
84.63%
60.47%
21
0.176
0.00%
0.03%
0.00%
84.78%
84.66%
60.47%
22
0.167
0.000
0.000
0.000
0.848
0.847
0.605
23
0.164
0.000
0.000
0.000
0.848
0.847
0.605
24
0.162
0.000
0.000
0.231
0.848
0.847
0.835
25
0.156
0.000
0.000
0.000
0.848
0.847
0.835
26
0.150
0.00%
0.01%
0.00%
84.78%
84.67%
83.53%
27
0.149
0.00%
0.00%
0.00%
84.78%
84.67%
83.53%
28
0.139
0.00%
0.00%
0.00%
84.78%
84.67%
83.53%
29
0.137
0.00%
0.00%
0.00%
84.78%
84.67%
83.53%
30
0.131
0.00%
0.00%
0.06%
84.78%
84.67%
83.59%
123
References
APPENDIX C
124
References
C.1
Transverse direction: Wall shear forces
Table C.0.1 Wall shear forces: comparison between DDBD hypotheses and DTHA response
Level
DDBD
Wall
shear
Vs,i
VAVERAGE
VAVERAGE
VAVERAGE
VAVERAGE
Case a
Case a
Case a
[-]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[%]
[%]
[%]
[%]
12
5496
4297
4177
4296
4177
21.8%
24.0%
21.8%
24.0%
11
6169
4297
4177
4296
4177
30.3%
32.3%
30.4%
32.3%
10
6842
3797
3913
3848
3913
44.5%
42.8%
43.8%
42.8%
9
7515
3797
3913
3848
3913
49.5%
47.9%
48.8%
47.9%
8
8188
4705
4919
5089
4919
42.5%
39.9%
37.8%
39.9%
7
8860
4705
4919
5089
4919
46.9%
44.5%
42.6%
44.5%
6
9533
5416
5354
5459
5355
43.2%
43.8%
42.7%
43.8%
5
10206
5416
5354
5459
5355
46.9%
47.5%
46.5%
47.5%
4
10879
7931
7950
8170
7950
27.1%
26.9%
24.9%
26.9%
3
11552
7931
7950
8170
7950
31.3%
31.2%
29.3%
31.2%
2
12225
8788
8852
8841
8852
28.1%
27.6%
27.7%
27.6%
1
12898
8788
8852
8841
8852
31.9%
31.4%
31.5%
31.4%
0
13739
13313
13406
13502
13406
3.1%
2.4%
1.7%
2.4%
C.2
DTHA Wall DTHA Wall DTHA Wall DTHA Wall
Percentige Percentige Percentige Percentige
Shear
Shear
Shear
Shear
Difference Difference Difference Difference
Case a
Case b
Case c
Case d
Case a
Transverse direction: Frame Shear forces in outer columns
Table C.0.2 Frame shear forces: comparison between DDBD hypotheses and DTHA response
DTHA
Frame
Shear
Case a
DTHA
Frame
Shear
Case b
DTHA
Frame
Shear
Case c
DTHA
Frame
Shear
Case d
VAVAREGE
VAVAREGE
VAVAREGE
VAVAREGE
[-]
Vs,i
[ KN ]
[ KN ]
[ KN ]
[ KN ]
12
461.1
506.35
429.05
11
461.1
621.25
10
461.1
621.25
9
461.1
8
461.1
7
DDBD
Frame
shear
Percentige
Difference
Percentige
Difference
Percentige
Difference
Percentige
Difference
[ KN ]
Case a
[%]
Case a
[%]
Case a
[%]
Case a
[%]
418.10
417.63
-8.9%
7.0%
9.3%
9.4%
534.60
520.40
521.03
-25.8%
-13.7%
-11.4%
-11.5%
534.60
520.40
521.03
-25.8%
-13.7%
-11.4%
-11.5%
610.50
526.73
513.80
514.33
-24.5%
-12.5%
-10.3%
-10.3%
623.32
537.86
525.20
526.12
-26.0%
-14.3%
-12.2%
-12.4%
461.1
634.00
543.60
531.80
532.08
-27.3%
-15.2%
-13.3%
-13.3%
6
461.1
635.81
556.24
546.60
546.14
-27.5%
-17.1%
-15.6%
-15.6%
5
461.1
635.81
556.24
546.60
546.14
-27.5%
-17.1%
-15.6%
-15.6%
4
461.1
638.49
559.84
548.80
549.38
-27.8%
-17.6%
-16.0%
-16.1%
3
461.1
638.49
559.84
548.80
549.38
-27.8%
-17.6%
-16.0%
-16.1%
2
461.1
662.24
592.02
579.20
579.05
-30.4%
-22.1%
-20.4%
-20.4%
1
461.1
329.95
265.25
253.90
253.69
28.4%
42.5%
44.9%
45.0%
0
461.1
329.95
265.25
253.90
253.69
28.4%
42.5%
44.9%
45.0%
Level
125
References
C.3
Transverse direction: Frame Moments in outer columns
Table C.0.3 Frame moments: comparison between DDBD hypotheses and DTHA response
Level
DDBD
Design
Moment
DTHA
Frame
Moment
Case a
DTHA
Frame
Moment
Case b
DTHA
Frame
Moment
Case c
DTHA
Frame
Moment
Case c
Percentige
Difference
Ms,i
[ KN m ]
MAVERAGE
MAVERAGE
MAVERAGE
MAVERAGE
[-]
[ KN m]
[ KN m]
[ KN m]
[ KN m]
Case a
[%]
12
11
10
9
8
7
6
5
4
3
2
1
0
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
1043.3
709.4
629.2
746.9
845.4
839.3
875.5
866.1
893.2
887.5
889.4
899.7
889.2
422.5
650.6
628.7
746.9
846.3
840.1
875.0
868.0
892.9
886.8
890.1
900.4
889.5
422.8
650.0
612.4
727.2
825.6
819.1
857.2
847.5
876.4
872.1
873.0
882.7
870.3
396.9
635.7
612.0
727.2
826.5
819.9
856.7
849.5
876.1
871.4
873.6
883.4
870.6
397.3
635.1
39.7%
28.4%
19.0%
19.6%
16.1%
17.0%
14.4%
14.9%
14.7%
13.8%
14.8%
59.5%
8.3%
C.4
Percentige
Difference
Percentige
difference
Percentige
difference
Case b
[%]
Case c
[%]
Case d
[%]
39.7%
28.4%
18.9%
19.5%
16.1%
16.8%
14.4%
15.0%
14.7%
13.7%
14.7%
59.5%
8.4%
41.3%
30.3%
20.9%
21.5%
17.8%
18.8%
16.0%
16.4%
16.3%
15.4%
16.6%
62.0%
10.4%
41.3%
30.3%
20.8%
21.4%
17.9%
18.6%
16.0%
16.5%
16.3%
15.3%
16.6%
61.9%
10.5%
Transverse direction: Frame Shear in inner columns
Table C.0.4 Frame shear forces: comparison between DDBD hypotheses and DTHA response
DDBD
Design
shear
DTHA
Frame
Shear
Case a
DTHA
Frame
Shear
Case b
DTHA
Frame
Shear
Case c
DTHA
Frame
Shear
Case d
Percentige
Difference
Percentige
Difference
Percentige
Difference
Percentige
Difference
Vs,i
VENVELOPE
VENVELOPE
VENVELOPE
VENVELOPE
Case a
Case b
Case c
Case d
[-]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[ KN ]
[%]
[%]
[%]
[%]
12
922.2
812.5
813.2
803.1
803.8
11.89%
11.81%
12.91%
12.83%
11
922.2
812.5
813.2
803.1
803.8
11.89%
11.81%
12.91%
12.83%
10
922.2
972.6
972.1
965.5
965.0
-5.19%
-5.14%
-4.49%
-4.44%
9
922.2
972.6
972.1
965.5
965.0
-5.19%
-5.14%
-4.49%
-4.44%
8
922.2
1003.6
1005.1
995.6
997.1
-8.11%
-8.25%
-7.38%
-7.52%
7
922.2
1003.6
1005.1
995.6
997.1
-8.11%
-8.25%
-7.38%
-7.52%
6
922.2
1014.5
1014.7
1010.9
1011.0
-9.10%
-9.12%
-8.78%
-8.79%
5
922.2
1014.5
1014.7
1010.9
1011.0
-9.10%
-9.12%
-8.78%
-8.79%
4
922.2
1042.1
1041.4
1040.1
1039.4
-11.51%
-11.45%
-11.34%
-11.28%
3
922.2
1042.1
1041.4
1040.1
1039.4
-11.51%
-11.45%
-11.34%
-11.28%
2
922.2
1396.0
1396.4
1376.7
1377.0
-33.94%
-33.96%
-33.01%
-33.03%
1
922.2
1396.0
1396.4
1376.7
1377.0
-33.94%
-33.96%
-33.01%
-33.03%
0
922.2
692.3
689.3
675.7
672.7
24.92%
25.25%
26.72%
27.05%
Level
126
References
C.5
Transverse direction: Frame Moment in inner columns
Table C.0.5 Frame moments: comparison between DDBD hypotheses and DTHA response
Level
DDBD
Moment
DTHA
Frame
Moment
Case a
DTHA
Frame
Moment
Case b
DTHA
Frame
Moment
Case c
DTHA
Frame
Moment
Case d
Percentige
Difference
Percentige
Difference
Percentige
difference
Percentige
difference
Ms,i
MAVERAGE
MAVERAGE
MAVERAGE
MAVERAGE
Case a
Case b
Case c
Case d
[-]
[ KN m ]
[ KN m]
[ KN m]
[ KN m]
[ KN m]
[%]
[%]
[%]
[%]
12
11
10
9
8
7
6
5
4
3
2
1
0
2087
2087
2087
2087
2087
2087
2087
2087
2087
2087
2087
2087
1419
1203
1409
1603
1534
1662
1571
1666
1617
1631
1721
1768
2703
2546
1188
1394
1588
1522
1643
1566
1651
1618
1625
1721
1757
2652
2480
1267
1470
1654
1597
1734
1641
1753
1687
1703
1797
1849
2864
2925
1191
1393
1589
1519
1641
1569
1652
1619
1623
1722
1756
2653
2480
42.3%
32.5%
23.2%
26.5%
20.4%
24.7%
20.2%
22.5%
21.9%
17.5%
15.3%
-22.8%
-44.3%
43.0%
33.2%
23.9%
27.1%
21.3%
25.0%
20.9%
22.5%
22.1%
17.5%
15.8%
-21.3%
-42.8%
39.3%
29.6%
20.7%
23.5%
16.9%
21.3%
16.0%
19.2%
18.4%
13.9%
11.4%
-27.1%
-51.5%
42.9%
33.3%
23.9%
27.2%
21.3%
24.8%
20.8%
22.4%
22.2%
17.5%
15.9%
-21.3%
-42.8%
127
References
Longitudinal direction: Displacement profiles
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
Relative Height ( hi/H )
Relative Height ( hi/H )
C.6
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.2
0.4
0.6
0.8
Lateral Displacement [ m ]
1.0
(b) Dynamic input : 1st set; Penalty combination: 2
0.0
(d)
0.2
0.4
0.6
0.8
Lateral Displacement [ m ]
1.0
Dynamic input :2nd set; Penalty combination:2
12
11
11
10
10
9
9
8
8
7
7
Level
Level
p
12
6
6
5
5
4
4
3
3
2
2
1
1
0
0.00
0.20
0.40
Design Displacement
st
0.60
0.80
0
0.00
1.00
Average DTHA Displacement
(b) Dynamic input : 1 set; Penalty combination: 2
0.20
0.40
Design Displacement
0.60
0.80
1.00
DTHA Displacement
Displacement [ m ]
(d)
Dynamic input :2nd set; Penalty combination:2
FigureC.0.1 Diplacement profiles: comparison between DDBD hypothesis and DTHA average response
128
References
Longitudinal direction: Drift profiles
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
C.7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0.0
1.0
2.0
3.0
0.0
4.0
1.0
(b) Dynamic input : 1st set; Penalty combination: 2
3.0
4.0
(d) Dynamic input :2nd set; Penalty combination:2
12
12
11
11
10
10
9
9
8
8
7
7
Level
Level
2.0
Storey Drift [ % ]
Storey Drift [ % ]
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
DTHA Drift
DDBD Drift
3
4
Code Drift
Interstorey Drift [ % ]
(b) Dynamic input : 1st set; Penalty combination: 2
0
0
1
DTHA Drift
2
3
DDBD Drift
4
Code Drift
Interstorey Drift [ % ]
(d) Dynamic input :2nd set; Penalty combination:2
FigureC.0.2 Drift profiles: comparison between DDBD hypothesis and DTHA average response
129